WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:03.430 align:middle line:90% [RUSTLING] 00:00:03.430 --> 00:00:06.370 align:middle line:90% [CLICKING] 00:00:06.370 --> 00:00:18.817 align:middle line:90% 00:00:18.817 --> 00:00:20.400 align:middle line:84% CASEY RODRIGUEZ: OK, so let's continue 00:00:20.400 --> 00:00:31.950 align:middle line:84% with our discussion of Fourier series from last time. 00:00:31.950 --> 00:00:36.790 align:middle line:90% 00:00:36.790 --> 00:00:45.540 align:middle line:84% For a given function in L2, we define the Fourier coefficient 00:00:45.540 --> 00:00:53.770 align:middle line:84% f hat of n to be the integral of 1 over 2 pi minus pi to pi 00:00:53.770 --> 00:01:01.000 align:middle line:84% f of t e to the minus int dt, which 00:01:01.000 --> 00:01:04.650 align:middle line:84% up to a factor of 1 over root t, is 00:01:04.650 --> 00:01:08.490 align:middle line:84% equal to the inner product of f with e 00:01:08.490 --> 00:01:15.066 align:middle line:84% to the int over square root of 2 pi in the Hilbert space L2, OK? 00:01:15.066 --> 00:01:17.790 align:middle line:90% 00:01:17.790 --> 00:01:23.970 align:middle line:90% And the question that we had-- 00:01:23.970 --> 00:01:32.620 align:middle line:84% so let me-- and we also had that the n-th partial sum 00:01:32.620 --> 00:01:34.300 align:middle line:84% for the Fourier series associated 00:01:34.300 --> 00:01:40.990 align:middle line:84% to f was given by sum from n equals minus n to n f hat of n 00:01:40.990 --> 00:01:43.330 align:middle line:90% e to the inx. 00:01:43.330 --> 00:01:49.310 align:middle line:84% And the question which we're trying to resolve 00:01:49.310 --> 00:02:03.220 align:middle line:84% is do we have for all f and L2 limit 00:02:03.220 --> 00:02:12.380 align:middle line:84% as n goes to infinity of f minus SN of f 2 equals 0, OK? 00:02:12.380 --> 00:02:17.400 align:middle line:90% 00:02:17.400 --> 00:02:17.900 align:middle line:90% All right? 00:02:17.900 --> 00:02:23.420 align:middle line:84% In other words, is f equal to its Fourier series, at least 00:02:23.420 --> 00:02:28.210 align:middle line:84% when we interpret equals as in this sense here? 00:02:28.210 --> 00:02:34.700 align:middle line:84% Now, based on what we've done for Hilbert spaces, 00:02:34.700 --> 00:02:40.310 align:middle line:84% this question is equivalent to the following statement. 00:02:40.310 --> 00:02:44.150 align:middle line:90% 00:02:44.150 --> 00:03:05.480 align:middle line:84% If f is an L2, and the Fourier coefficients are all 0, 00:03:05.480 --> 00:03:07.910 align:middle line:84% does this imply f equals 0, right? 00:03:07.910 --> 00:03:11.270 align:middle line:90% So this question is-- 00:03:11.270 --> 00:03:13.160 align:middle line:84% by what we've done for Hilbert spaces, 00:03:13.160 --> 00:03:15.105 align:middle line:90% big L2 is a Hilbert space. 00:03:15.105 --> 00:03:17.480 align:middle line:84% This question here is equivalent to this statement, which 00:03:17.480 --> 00:03:25.390 align:middle line:84% is that the collection of orthonormal vectors and big L2 00:03:25.390 --> 00:03:28.750 align:middle line:84% consisting of the exponentials divided by square root of 2-- 00:03:28.750 --> 00:03:31.060 align:middle line:84% is this a maximal orthonormal subset? 00:03:31.060 --> 00:03:34.390 align:middle line:84% Or as we were using the terminology 00:03:34.390 --> 00:03:38.510 align:middle line:84% we had from last time, does that form an orthonormal basis? 00:03:38.510 --> 00:03:41.230 align:middle line:84% So this is the statement that we're 00:03:41.230 --> 00:03:43.285 align:middle line:90% going to prove this class. 00:03:43.285 --> 00:03:52.110 align:middle line:90% 00:03:52.110 --> 00:03:55.860 align:middle line:84% And we're going to proceed via Fejer's method, 00:03:55.860 --> 00:04:02.130 align:middle line:84% if you like to give it a name, where what we did last time 00:04:02.130 --> 00:04:02.820 align:middle line:90% was-- 00:04:02.820 --> 00:04:06.030 align:middle line:84% when we recall, we had the Cesaro Fourier 00:04:06.030 --> 00:04:12.760 align:middle line:84% mean we defined to be the average 00:04:12.760 --> 00:04:22.950 align:middle line:84% of the first n partial sums with the hope 00:04:22.950 --> 00:04:26.190 align:middle line:84% that somehow this behaves a little bit better 00:04:26.190 --> 00:04:28.620 align:middle line:84% than the partial sums, because that's the thing we're 00:04:28.620 --> 00:04:29.290 align:middle line:90% trying to study. 00:04:29.290 --> 00:04:30.415 align:middle line:90% And that's a hard question. 00:04:30.415 --> 00:04:34.770 align:middle line:84% And typically, means of sequences 00:04:34.770 --> 00:04:37.650 align:middle line:84% might behave better than the sequences themselves. 00:04:37.650 --> 00:04:39.850 align:middle line:84% But if the original sequence converges, 00:04:39.850 --> 00:04:40.900 align:middle line:90% then the means converge. 00:04:40.900 --> 00:04:42.960 align:middle line:84% So we should expect this to converge 00:04:42.960 --> 00:04:50.350 align:middle line:84% to f but hopefully faster or have better, more 00:04:50.350 --> 00:04:53.620 align:middle line:84% recognizable properties than just studying 00:04:53.620 --> 00:04:57.440 align:middle line:90% the partial sums directly. 00:04:57.440 --> 00:05:00.800 align:middle line:84% And we'll get to-- in the next statement, 00:05:00.800 --> 00:05:05.750 align:middle line:84% it's a little bit clearer why the Cesaro Fourier means 00:05:05.750 --> 00:05:08.600 align:middle line:90% converge to f. 00:05:08.600 --> 00:05:10.880 align:middle line:90% And so what our goal-- 00:05:10.880 --> 00:05:15.580 align:middle line:90% 00:05:15.580 --> 00:05:24.580 align:middle line:84% what we're going to show is that if f is an L2, 00:05:24.580 --> 00:05:34.860 align:middle line:84% then the Cesaro Fourier means converts to 0, 00:05:34.860 --> 00:05:40.887 align:middle line:84% I mean converts to f as n goes to infinity. 00:05:40.887 --> 00:05:42.290 align:middle line:90% OK? 00:05:42.290 --> 00:05:44.350 align:middle line:84% And so once we've proven that, then that 00:05:44.350 --> 00:05:47.848 align:middle line:84% gives us what we want in the yellow box, right? 00:05:47.848 --> 00:05:49.390 align:middle line:84% That proves what's in the yellow box, 00:05:49.390 --> 00:05:54.640 align:middle line:84% because let's take f and L2 with 48 coefficients all 0. 00:05:54.640 --> 00:05:57.880 align:middle line:84% Then, all of the partial sums will be 0. 00:05:57.880 --> 00:06:00.280 align:middle line:84% Then, all of the means will be 0. 00:06:00.280 --> 00:06:03.610 align:middle line:84% And since the means converge to f, that proves f is 0. 00:06:03.610 --> 00:06:05.290 align:middle line:84% And we get what's in the yellow box. 00:06:05.290 --> 00:06:10.360 align:middle line:84% And therefore, the partial the Fourier sums 00:06:10.360 --> 00:06:16.100 align:middle line:84% converts to f as capital N goes to infinity in L2, all right? 00:06:16.100 --> 00:06:17.810 align:middle line:84% And then, once we prove that, I'll 00:06:17.810 --> 00:06:21.030 align:middle line:84% make a couple of comments about other types of questions 00:06:21.030 --> 00:06:26.190 align:middle line:84% one can ask and what you can do, or a brief comment. 00:06:26.190 --> 00:06:31.290 align:middle line:84% So this is our goal for this lecture. 00:06:31.290 --> 00:06:33.940 align:middle line:90% 00:06:33.940 --> 00:06:38.270 align:middle line:84% And we should be able to get through it. 00:06:38.270 --> 00:06:47.390 align:middle line:84% So let me first rewrite the Cesaro Fourier means 00:06:47.390 --> 00:06:48.920 align:middle line:90% slightly differently. 00:06:48.920 --> 00:06:50.690 align:middle line:84% How we did in the previous lecture 00:06:50.690 --> 00:06:54.230 align:middle line:84% for the partial Fourier sums-- we wrote them 00:06:54.230 --> 00:06:56.870 align:middle line:90% as what's called a convolution. 00:06:56.870 --> 00:06:59.780 align:middle line:84% I haven't defined convolution-- but an integral 00:06:59.780 --> 00:07:05.470 align:middle line:84% of a function depending on x and t times f of t dt. 00:07:05.470 --> 00:07:10.000 align:middle line:84% And we're going to do the same now for the Cesaro means. 00:07:10.000 --> 00:07:12.580 align:middle line:90% And we'll see here in what-- 00:07:12.580 --> 00:07:15.870 align:middle line:90% 00:07:15.870 --> 00:07:18.370 align:middle line:84% it's a little bit more clear, although I didn't talk so much 00:07:18.370 --> 00:07:20.703 align:middle line:84% about the Dirichlet kernel that appears for these guys-- 00:07:20.703 --> 00:07:24.280 align:middle line:84% but why the Cesaro means converge to f. 00:07:24.280 --> 00:07:27.220 align:middle line:90% 00:07:27.220 --> 00:07:39.780 align:middle line:84% OK, all right, so the statement is for all f in L2 minus pi 00:07:39.780 --> 00:07:53.560 align:middle line:84% to pi, we have that the n-th Cesaro 00:07:53.560 --> 00:07:56.450 align:middle line:84% mean of f, which is our Fourier mean, 00:07:56.450 --> 00:07:58.930 align:middle line:84% I can write as the integral from minus pi 00:07:58.930 --> 00:08:07.060 align:middle line:84% to pi of a function kn of x minus t times f of t dt. 00:08:07.060 --> 00:08:08.740 align:middle line:84% So remember, for the partial sums 00:08:08.740 --> 00:08:11.080 align:middle line:84% s sub n, we could write it as d sub n, 00:08:11.080 --> 00:08:20.610 align:middle line:84% where d denoted a Dirichlet kernel, where 00:08:20.610 --> 00:08:30.110 align:middle line:84% kn of x-- this is equal to n plus 1 over 2 pi, 00:08:30.110 --> 00:08:36.770 align:middle line:84% and then 1 over 2 pi times n plus 1 times sine n 00:08:36.770 --> 00:08:44.510 align:middle line:84% plus 1 over 2x over sine x over 2 squared. 00:08:44.510 --> 00:08:46.220 align:middle line:90% And this holds that x equals 0. 00:08:46.220 --> 00:08:49.322 align:middle line:90% This one holds at x0 equal to 0. 00:08:49.322 --> 00:08:51.150 align:middle line:90% OK? 00:08:51.150 --> 00:09:01.800 align:middle line:84% And this thing, we call Fejer's or the Fejer kernel. 00:09:01.800 --> 00:09:04.760 align:middle line:90% So all right? 00:09:04.760 --> 00:09:07.340 align:middle line:90% 00:09:07.340 --> 00:09:12.980 align:middle line:84% And now, let me just list off a few properties 00:09:12.980 --> 00:09:14.090 align:middle line:90% that we'll get from this. 00:09:14.090 --> 00:09:20.690 align:middle line:90% 00:09:20.690 --> 00:09:23.360 align:middle line:84% Moreover, we have the following properties. 00:09:23.360 --> 00:09:30.156 align:middle line:90% 1, kn is non-negative. 00:09:30.156 --> 00:09:32.880 align:middle line:90% 00:09:32.880 --> 00:09:37.950 align:middle line:84% kn of x is equal to kn of minus x. 00:09:37.950 --> 00:09:40.690 align:middle line:90% It's even. 00:09:40.690 --> 00:09:51.280 align:middle line:90% And kn is 2 pi periodic. 00:09:51.280 --> 00:09:53.640 align:middle line:84% The second is that the integral of k 00:09:53.640 --> 00:09:58.470 align:middle line:90% sub n of x from minus pi to pi-- 00:09:58.470 --> 00:10:01.960 align:middle line:90% 00:10:01.960 --> 00:10:05.688 align:middle line:90% dx, or let's make this t-- 00:10:05.688 --> 00:10:08.460 align:middle line:90% dt equals 1. 00:10:08.460 --> 00:10:17.810 align:middle line:84% And the third is if delta is a positive number less than pi, 00:10:17.810 --> 00:10:29.810 align:middle line:84% less than or equal to pi, then for all x with absolute value 00:10:29.810 --> 00:10:35.110 align:middle line:84% bigger than or equal to delta and less than or equal to pi, 00:10:35.110 --> 00:10:43.335 align:middle line:84% we have that kn of x, which is equal-- 00:10:43.335 --> 00:10:44.710 align:middle line:84% I don't need the absolute values, 00:10:44.710 --> 00:10:49.510 align:middle line:84% because it's non-negative-- is less than or equal to 1 over 2 00:10:49.510 --> 00:10:59.790 align:middle line:84% pi over n plus 1 times sine squared delta over 2, OK? 00:10:59.790 --> 00:11:03.990 align:middle line:90% So OK. 00:11:03.990 --> 00:11:06.540 align:middle line:90% So let's prove this theorem. 00:11:06.540 --> 00:11:08.820 align:middle line:84% And then, I'm going to say a few comments about-- 00:11:08.820 --> 00:11:11.110 align:middle line:84% well, since I have these properties right here, 00:11:11.110 --> 00:11:14.550 align:middle line:84% let me go ahead and make a few comments before we prove it. 00:11:14.550 --> 00:11:20.640 align:middle line:84% What does that mean kn looks like? 00:11:20.640 --> 00:11:27.730 align:middle line:90% Let me draw 0 pi minus pi. 00:11:27.730 --> 00:11:35.590 align:middle line:90% 00:11:35.590 --> 00:11:37.060 align:middle line:90% So kn is non-negative. 00:11:37.060 --> 00:11:38.950 align:middle line:90% It's even. 00:11:38.950 --> 00:11:42.550 align:middle line:84% And away from a small neighborhood, 00:11:42.550 --> 00:11:46.130 align:middle line:84% it's quite small if capital N is very big. 00:11:46.130 --> 00:11:51.580 align:middle line:84% So what it's looking like is maybe the first one-- 00:11:51.580 --> 00:11:54.385 align:middle line:90% and it's large at the origin. 00:11:54.385 --> 00:12:04.830 align:middle line:90% 00:12:04.830 --> 00:12:09.800 align:middle line:84% OK, so maybe that's n equals let's say 1. 00:12:09.800 --> 00:12:14.390 align:middle line:84% And then, let's say this is delta, and then minus delta. 00:12:14.390 --> 00:12:18.260 align:middle line:84% If I were to now look at, let's say, n equals-- 00:12:18.260 --> 00:12:25.420 align:middle line:84% I don't know-- a billion, it looks more like something 00:12:25.420 --> 00:12:29.500 align:middle line:84% that's very concentrated at the origin, 00:12:29.500 --> 00:12:37.630 align:middle line:84% but in such a way that the area underneath the graph-- 00:12:37.630 --> 00:12:43.240 align:middle line:84% so the integral-- the area equals 1, OK? 00:12:43.240 --> 00:12:48.190 align:middle line:84% And the same with what I drew in white, because white 00:12:48.190 --> 00:12:49.720 align:middle line:90% was supposed to be n equals 1. 00:12:49.720 --> 00:12:52.030 align:middle line:84% Yellow was supposed to be n equals-- 00:12:52.030 --> 00:12:53.890 align:middle line:90% I don't know-- 1,000. 00:12:53.890 --> 00:12:58.480 align:middle line:90% The area is always 1, OK? 00:12:58.480 --> 00:13:07.120 align:middle line:84% So this is telling you that if I look at sigma n of f-- 00:13:07.120 --> 00:13:09.140 align:middle line:90% so this is just some remarks. 00:13:09.140 --> 00:13:12.520 align:middle line:84% This is not to be taken completely literally. 00:13:12.520 --> 00:13:14.800 align:middle line:84% This is just the intuition on why 00:13:14.800 --> 00:13:20.020 align:middle line:84% we believe that the Cesaro means converge to f. 00:13:20.020 --> 00:13:25.110 align:middle line:84% And I'll say how this picture differs 00:13:25.110 --> 00:13:27.210 align:middle line:90% from if we looked at just SN. 00:13:27.210 --> 00:13:38.450 align:middle line:84% So this means that sigma n of f is, in fact-- 00:13:38.450 --> 00:13:41.970 align:middle line:90% 00:13:41.970 --> 00:13:43.860 align:middle line:84% so remember, we're going to get, in the end, 00:13:43.860 --> 00:13:55.460 align:middle line:84% that this is equal to kn of x minus t f of t dt. 00:13:55.460 --> 00:14:04.580 align:middle line:84% Now, kn is very concentrated near where t equals x, OK? 00:14:04.580 --> 00:14:08.330 align:middle line:84% So based on the picture, as n gets very large, 00:14:08.330 --> 00:14:11.690 align:middle line:84% this thing is getting more and more concentrated near where 00:14:11.690 --> 00:14:13.580 align:middle line:90% n equals x, OK? 00:14:13.580 --> 00:14:18.560 align:middle line:84% Now, and therefore, at least for let's say very nice 00:14:18.560 --> 00:14:23.840 align:middle line:84% f, if this thing is concentrated near where t equals x, then f 00:14:23.840 --> 00:14:28.370 align:middle line:84% of t will be approximately f of x. 00:14:28.370 --> 00:14:31.398 align:middle line:84% So f of x comes out of the integral 00:14:31.398 --> 00:14:32.690 align:middle line:90% because this is an integral dt. 00:14:32.690 --> 00:14:37.520 align:middle line:90% 00:14:37.520 --> 00:14:39.800 align:middle line:84% So since this thing is concentrated at-- 00:14:39.800 --> 00:14:44.050 align:middle line:90% 00:14:44.050 --> 00:14:47.850 align:middle line:84% and because the area underneath the curve is always 1, 00:14:47.850 --> 00:14:51.750 align:middle line:84% this integral is always equal to the integral of kn over any. 00:14:51.750 --> 00:14:53.730 align:middle line:90% So kn is 2 pi periodic. 00:14:53.730 --> 00:15:01.050 align:middle line:84% This integral is equal to the same integral over any 2 pi 00:15:01.050 --> 00:15:05.280 align:middle line:84% periodic interval, which means I could put here-- 00:15:05.280 --> 00:15:07.350 align:middle line:84% I could add an x to both top and bottom, 00:15:07.350 --> 00:15:10.215 align:middle line:84% and therefore change variables to get this 00:15:10.215 --> 00:15:14.910 align:middle line:90% is kn of t dt, which equals-- 00:15:14.910 --> 00:15:17.460 align:middle line:84% because the integral is 1, I would get something 00:15:17.460 --> 00:15:18.630 align:middle line:90% like f of x, OK? 00:15:18.630 --> 00:15:27.190 align:middle line:84% So this is a heuristic reason on why 00:15:27.190 --> 00:15:34.520 align:middle line:84% one should expect the Cesaro means to converge to f. 00:15:34.520 --> 00:15:35.020 align:middle line:90% OK? 00:15:35.020 --> 00:16:06.170 align:middle line:90% 00:16:06.170 --> 00:16:08.810 align:middle line:84% If you look back at the kernel that we 00:16:08.810 --> 00:16:15.490 align:middle line:84% had for the partial sums, it had some 00:16:15.490 --> 00:16:18.730 align:middle line:90% of the same-ish properties. 00:16:18.730 --> 00:16:26.410 align:middle line:84% It was 2 pi periodic, and also even. 00:16:26.410 --> 00:16:28.610 align:middle line:90% The integral was 1. 00:16:28.610 --> 00:16:36.590 align:middle line:90% And it did decay away from 0. 00:16:36.590 --> 00:16:39.927 align:middle line:90% However, it's non-negative. 00:16:39.927 --> 00:16:41.510 align:middle line:84% I'm talking about the Dirichlet kernel 00:16:41.510 --> 00:16:45.425 align:middle line:84% dn, which if you look back in your notes, was sine of plus n 00:16:45.425 --> 00:16:48.470 align:middle line:84% plus 1/2 times x over sine x over 2 00:16:48.470 --> 00:16:50.660 align:middle line:90% with a constant out in front. 00:16:50.660 --> 00:16:54.200 align:middle line:84% And that little difference, the fact that this kernel is 00:16:54.200 --> 00:16:56.810 align:middle line:84% non-negative-- and the Dirichlet kernel is not-- 00:16:56.810 --> 00:16:58.320 align:middle line:90% makes a big difference. 00:16:58.320 --> 00:17:00.200 align:middle line:84% So although this heuristic argument-- 00:17:00.200 --> 00:17:01.970 align:middle line:90% maybe you don't see it there-- 00:17:01.970 --> 00:17:06.510 align:middle line:84% in the actual proof itself, that oscillation-- 00:17:06.510 --> 00:17:08.520 align:middle line:84% and what I mean by isolation is the fact 00:17:08.520 --> 00:17:11.520 align:middle line:84% that dn actually does oscillate between negative and positive 00:17:11.520 --> 00:17:13.530 align:middle line:90% values-- 00:17:13.530 --> 00:17:15.450 align:middle line:84% this bit of oscillation is actually 00:17:15.450 --> 00:17:20.310 align:middle line:84% what you can use to build up a continuous function whose 00:17:20.310 --> 00:17:23.640 align:middle line:84% partial sums do not converge to that continuous function 00:17:23.640 --> 00:17:25.020 align:middle line:90% at a point, OK? 00:17:25.020 --> 00:17:30.270 align:middle line:84% But as we'll see for the Cesaro means, 00:17:30.270 --> 00:17:34.110 align:middle line:84% the Cesaro Fourier means, basically, pick a space. 00:17:34.110 --> 00:17:36.720 align:middle line:84% And the Cesaro sums or Cesaro means 00:17:36.720 --> 00:17:40.350 align:middle line:84% converge to the function in whatever space 00:17:40.350 --> 00:17:42.870 align:middle line:90% you're talking about. 00:17:42.870 --> 00:17:45.120 align:middle line:84% And I'll say a little bit more about that in a minute. 00:17:45.120 --> 00:17:45.890 align:middle line:90% But OK. 00:17:45.890 --> 00:17:49.220 align:middle line:84% So let's prove the theorem that the Cesaro means are 00:17:49.220 --> 00:17:52.580 align:middle line:84% written in this way, and the kernel has these three 00:17:52.580 --> 00:17:53.390 align:middle line:90% properties. 00:17:53.390 --> 00:18:15.610 align:middle line:90% 00:18:15.610 --> 00:18:23.820 align:middle line:84% So let me recall that we have SN of x-- 00:18:23.820 --> 00:18:26.400 align:middle line:90% or let's put a k there-- 00:18:26.400 --> 00:18:29.610 align:middle line:84% this is equal to, as we wrote last time, minus pi 00:18:29.610 --> 00:18:40.080 align:middle line:84% to pi DK of x minus t f of t dt, where 00:18:40.080 --> 00:18:57.980 align:middle line:84% DK of, say, t was, from last time, equal to 2n 00:18:57.980 --> 00:19:06.620 align:middle line:84% plus 1 over 2 pi at t equals 0 and sine n 00:19:06.620 --> 00:19:16.430 align:middle line:84% plus 1/2 t over sine t over 2, and then with 1 over 2 00:19:16.430 --> 00:19:18.180 align:middle line:90% pi out in front, I believe. 00:19:18.180 --> 00:19:21.530 align:middle line:84% Let me make sure I got the right exponent. 00:19:21.530 --> 00:19:24.800 align:middle line:90% 00:19:24.800 --> 00:19:25.300 align:middle line:90% Right. 00:19:25.300 --> 00:19:28.140 align:middle line:90% 00:19:28.140 --> 00:19:31.089 align:middle line:90% For t not equal to 0. 00:19:31.089 --> 00:19:33.810 align:middle line:90% OK? 00:19:33.810 --> 00:19:34.880 align:middle line:90% Oh, and this should be k. 00:19:34.880 --> 00:19:39.480 align:middle line:90% 00:19:39.480 --> 00:19:49.130 align:middle line:84% OK, so using this, we have that the Cesaro 00:19:49.130 --> 00:20:01.500 align:middle line:84% sum of x-- this is equal to 1 over n plus 1, sum from k 00:20:01.500 --> 00:20:08.310 align:middle line:84% equals 0 to n, the mean of the first and partial sums. 00:20:08.310 --> 00:20:09.900 align:middle line:90% And this is equal to-- 00:20:09.900 --> 00:20:12.220 align:middle line:90% now, sk f of x is equal to this. 00:20:12.220 --> 00:20:17.910 align:middle line:84% So I can write this as integral from minus pi to pi of 1 over n 00:20:17.910 --> 00:20:29.594 align:middle line:84% plus 1 sum from k equals 0 to n of DK x minus t f of t dt. 00:20:29.594 --> 00:20:33.010 align:middle line:90% 00:20:33.010 --> 00:20:41.680 align:middle line:84% And so this here is kn of x minus t. 00:20:41.680 --> 00:20:42.180 align:middle line:90% All right? 00:20:42.180 --> 00:20:45.750 align:middle line:84% So now, I'm just going to verify that kn of x takes 00:20:45.750 --> 00:20:48.285 align:middle line:90% that form that we had before. 00:20:48.285 --> 00:20:52.800 align:middle line:90% 00:20:52.800 --> 00:20:57.410 align:middle line:84% And kn of x-- this is equal to 1 over n 00:20:57.410 --> 00:21:07.400 align:middle line:84% plus 1 sum from k equals 0 to n of DK of x. 00:21:07.400 --> 00:21:11.345 align:middle line:84% And let's go to the next half board. 00:21:11.345 --> 00:21:19.070 align:middle line:90% 00:21:19.070 --> 00:21:32.330 align:middle line:84% So I can write this as 1 over 2 pi n plus 1 and times-- 00:21:32.330 --> 00:21:36.050 align:middle line:84% so I will look at the case that x is non-0. 00:21:36.050 --> 00:21:39.740 align:middle line:84% x equals 0 is, you'll get what you get. 00:21:39.740 --> 00:21:43.190 align:middle line:84% But let's look at x not equal to 0. 00:21:43.190 --> 00:21:47.630 align:middle line:84% So then, I plug in this formula here 00:21:47.630 --> 00:21:53.360 align:middle line:90% and pull out a sine t over 2-- 00:21:53.360 --> 00:21:57.800 align:middle line:84% or sine x over 2 squared on the bottom. 00:21:57.800 --> 00:22:09.350 align:middle line:84% And then, I get k equals 0 to n of sine x over 2 times 00:22:09.350 --> 00:22:14.744 align:middle line:90% sine n plus 1/2 x, OK? 00:22:14.744 --> 00:22:17.410 align:middle line:90% 00:22:17.410 --> 00:22:23.780 align:middle line:84% And because I feel like it, let me put a 2 here and a 2 here. 00:22:23.780 --> 00:22:24.860 align:middle line:90% Why do I feel like it? 00:22:24.860 --> 00:22:28.690 align:middle line:84% Well, it's because if I have 2 times sine of a sine b, 00:22:28.690 --> 00:22:32.230 align:middle line:84% I can write that as using my angle sum 00:22:32.230 --> 00:22:34.750 align:middle line:90% formulas from trigonometry. 00:22:34.750 --> 00:22:37.040 align:middle line:84% You wondered why those would be useful. 00:22:37.040 --> 00:22:45.530 align:middle line:84% Well, here they are appearing in the advanced MIT class. 00:22:45.530 --> 00:22:52.910 align:middle line:84% You can write this as sum from k equals 0 to n of cosine n 00:22:52.910 --> 00:22:58.590 align:middle line:90% x minus cosine n plus 1x. 00:22:58.590 --> 00:23:03.054 align:middle line:90% 00:23:03.054 --> 00:23:04.610 align:middle line:84% Let me make sure I got that right. 00:23:04.610 --> 00:23:09.150 align:middle line:90% 00:23:09.150 --> 00:23:10.410 align:middle line:90% Or this should be k. 00:23:10.410 --> 00:23:11.690 align:middle line:90% I'm sorry. 00:23:11.690 --> 00:23:12.690 align:middle line:90% That should have been k. 00:23:12.690 --> 00:23:15.520 align:middle line:90% 00:23:15.520 --> 00:23:19.200 align:middle line:90% k, k, all right? 00:23:19.200 --> 00:23:21.210 align:middle line:84% Now this, is a telescoping sum, right? 00:23:21.210 --> 00:23:23.340 align:middle line:90% I have a sum of cosine kx. 00:23:23.340 --> 00:23:25.770 align:middle line:90% I have a cosine k plus 1x. 00:23:25.770 --> 00:23:27.525 align:middle line:90% So this is equal to-- 00:23:27.525 --> 00:23:35.310 align:middle line:90% 00:23:35.310 --> 00:23:36.890 align:middle line:90% so let's just write this out. 00:23:36.890 --> 00:23:41.510 align:middle line:90% 00:23:41.510 --> 00:23:44.960 align:middle line:84% And let me just indicate why this is a telescoping sum. 00:23:44.960 --> 00:23:54.850 align:middle line:84% We get cosine 0x minus cosine 1x plus cosine 1x 00:23:54.850 --> 00:24:00.280 align:middle line:84% minus cosine 2x dot dot dot plus the last one, which 00:24:00.280 --> 00:24:06.425 align:middle line:84% is cosine nx minus cosine n plus 1x. 00:24:06.425 --> 00:24:09.260 align:middle line:90% 00:24:09.260 --> 00:24:11.420 align:middle line:90% And OK, so this telescopes. 00:24:11.420 --> 00:24:13.340 align:middle line:90% That cancels with this. 00:24:13.340 --> 00:24:15.740 align:middle line:90% That will cancel with so on. 00:24:15.740 --> 00:24:17.570 align:middle line:90% And that last one will cancel. 00:24:17.570 --> 00:24:20.060 align:middle line:84% So all that we're left with is this one 00:24:20.060 --> 00:24:23.930 align:middle line:84% minus this one divided by this 2 that I have right there. 00:24:23.930 --> 00:24:31.580 align:middle line:84% And I get 1 over 2 pi n plus 1 times 1 over sine squared 00:24:31.580 --> 00:24:41.990 align:middle line:84% x over 2 times 1 minus cosine n plus 1 x over 2. 00:24:41.990 --> 00:24:48.340 align:middle line:84% And again, using a trig formula-- 00:24:48.340 --> 00:24:53.320 align:middle line:84% 1 minus cosine 2a equals sine squared-- 00:24:53.320 --> 00:24:55.480 align:middle line:84% divided by 2 is equal to sine squared a. 00:24:55.480 --> 00:25:02.380 align:middle line:84% So I get this is equal to 1 over 2 pi n plus 1 times sine 00:25:02.380 --> 00:25:12.610 align:middle line:84% squared n plus 1 over 2x divided by sine squared x over 2, OK? 00:25:12.610 --> 00:25:17.550 align:middle line:90% 00:25:17.550 --> 00:25:21.570 align:middle line:84% So that verifies the formula for the Fejer kernel. 00:25:21.570 --> 00:25:24.180 align:middle line:84% What about the properties that we have there? 00:25:24.180 --> 00:25:29.400 align:middle line:84% These properties-- at least the first two-- 00:25:29.400 --> 00:25:32.800 align:middle line:84% follow directly from this formula and the definition. 00:25:32.800 --> 00:25:43.930 align:middle line:90% So 1, follows immediately. 00:25:43.930 --> 00:25:45.730 align:middle line:90% This is clearly non-negative. 00:25:45.730 --> 00:25:50.530 align:middle line:84% It's even, taking x to minus x does not change this, 00:25:50.530 --> 00:25:51.640 align:middle line:90% because we have squares. 00:25:51.640 --> 00:25:55.270 align:middle line:84% And also because of the squares, it's 2 pi periodic 00:25:55.270 --> 00:25:58.822 align:middle line:90% rather than 4 pi periodic, OK? 00:25:58.822 --> 00:26:01.740 align:middle line:90% 00:26:01.740 --> 00:26:06.480 align:middle line:90% OK, so that's 1. 00:26:06.480 --> 00:26:16.480 align:middle line:84% For 2, we note that if we take the integral 00:26:16.480 --> 00:26:24.370 align:middle line:84% for minus pi to pi of the Dirichlet kernel, this is-- 00:26:24.370 --> 00:26:28.840 align:middle line:84% OK, we had a formula for the Dirichlet kernel, but remember, 00:26:28.840 --> 00:26:29.905 align:middle line:90% this is nothing but-- 00:26:29.905 --> 00:26:32.740 align:middle line:90% 00:26:32.740 --> 00:26:38.440 align:middle line:84% this was defined to be the sum from n equals minus k to k of e 00:26:38.440 --> 00:26:42.880 align:middle line:90% to the int dt, OK? 00:26:42.880 --> 00:26:49.420 align:middle line:84% Now, e to the int when n is not equal to 0 is 2 pi periodic. 00:26:49.420 --> 00:26:54.860 align:middle line:84% And when I integrate it from minus pi to pi, 00:26:54.860 --> 00:26:59.620 align:middle line:84% the integral from minus pi to pi of this 2 pi periodic thing-- 00:26:59.620 --> 00:27:00.670 align:middle line:90% you can just check. 00:27:00.670 --> 00:27:07.750 align:middle line:84% It's the integral of sine, nt, and cosine nt over its period. 00:27:07.750 --> 00:27:09.040 align:middle line:90% That's going to give me 0. 00:27:09.040 --> 00:27:12.340 align:middle line:84% So all I pick up is when n equals 0, right? 00:27:12.340 --> 00:27:16.850 align:middle line:84% And so that's equal to just the n equals 0 term. 00:27:16.850 --> 00:27:19.965 align:middle line:90% So that gives me 1, OK? 00:27:19.965 --> 00:27:27.330 align:middle line:90% 00:27:27.330 --> 00:27:30.570 align:middle line:84% So since the integral of each kernel 00:27:30.570 --> 00:27:37.530 align:middle line:84% is 1, then the integral of the Fejer kernel-- 00:27:37.530 --> 00:27:41.180 align:middle line:90% 00:27:41.180 --> 00:27:47.265 align:middle line:84% which remember, this is equal to the average of the Dirichlet 00:27:47.265 --> 00:27:47.765 align:middle line:90% kernels. 00:27:47.765 --> 00:28:04.390 align:middle line:90% 00:28:04.390 --> 00:28:08.560 align:middle line:84% And each of these is 1 sum from k equals 0 to n 1. 00:28:08.560 --> 00:28:10.600 align:middle line:84% I get n plus 1 divided by n plus 1. 00:28:10.600 --> 00:28:13.210 align:middle line:90% I get 1, OK? 00:28:13.210 --> 00:28:14.290 align:middle line:90% So that gives me 2. 00:28:14.290 --> 00:28:30.023 align:middle line:84% And for the third property, we have-- 00:28:30.023 --> 00:28:30.690 align:middle line:90% what do we have? 00:28:30.690 --> 00:28:41.220 align:middle line:90% 00:28:41.220 --> 00:28:47.550 align:middle line:84% Then, the function sine squared x over 2-- 00:28:47.550 --> 00:28:50.580 align:middle line:90% what does it look like? 00:28:50.580 --> 00:28:51.740 align:middle line:90% This is increasing. 00:28:51.740 --> 00:28:55.872 align:middle line:90% 00:28:55.872 --> 00:29:13.930 align:middle line:84% Or I should say it's even and increasing on 0 to pi. 00:29:13.930 --> 00:29:19.820 align:middle line:84% So what it looks like is sine squared x over 2. 00:29:19.820 --> 00:29:25.220 align:middle line:84% So there's pi minus pi sine squared. 00:29:25.220 --> 00:29:34.220 align:middle line:90% Looks like it goes up to 1. 00:29:34.220 --> 00:29:42.390 align:middle line:84% So if I'm looking at all x outside of-- 00:29:42.390 --> 00:29:44.400 align:middle line:90% so in that shaded region-- 00:29:44.400 --> 00:30:04.600 align:middle line:84% then, if x is outside of this delta region, 00:30:04.600 --> 00:30:08.710 align:middle line:84% then I get that sine squared x over 2 00:30:08.710 --> 00:30:18.660 align:middle line:84% is going to be bigger than or equal to whatever, 00:30:18.660 --> 00:30:21.660 align:middle line:84% so it sits above the value that I get here, which is sine 00:30:21.660 --> 00:30:23.355 align:middle line:90% squared delta over 2. 00:30:23.355 --> 00:30:26.820 align:middle line:90% 00:30:26.820 --> 00:30:33.250 align:middle line:84% And therefore, I get that kn of x, 00:30:33.250 --> 00:30:35.490 align:middle line:84% which is equal to its absolute value, 00:30:35.490 --> 00:30:47.340 align:middle line:84% is less than or equal to 1 over 2 pi n plus 1 sine squared 00:30:47.340 --> 00:30:50.610 align:middle line:90% n plus 1 over 2x over-- 00:30:50.610 --> 00:30:53.790 align:middle line:84% I had sine squared x over 2, but since sine squared x over 2 00:30:53.790 --> 00:30:57.150 align:middle line:84% is bigger than or equal to sine squared delta over 2, taking 1 00:30:57.150 --> 00:30:58.830 align:middle line:90% over reverses the inequalities. 00:30:58.830 --> 00:31:01.830 align:middle line:84% And I get sine squared delta over 2 here. 00:31:01.830 --> 00:31:04.500 align:middle line:84% Sine of anything is always bounded above by 1. 00:31:04.500 --> 00:31:09.510 align:middle line:84% So I get this is less than or equal to 1 over 2 pi n plus 1 00:31:09.510 --> 00:31:14.250 align:middle line:90% sine squared delta over 2, OK? 00:31:14.250 --> 00:31:30.050 align:middle line:90% 00:31:30.050 --> 00:31:37.350 align:middle line:84% So for the moment, let me put this absolute value there. 00:31:37.350 --> 00:31:42.002 align:middle line:84% I'm not doing it because I think it looks better. 00:31:42.002 --> 00:31:44.210 align:middle line:84% It's because I'm going to make a comment in a minute. 00:31:44.210 --> 00:31:46.820 align:middle line:90% 00:31:46.820 --> 00:31:58.455 align:middle line:84% OK, let me just make a small comment. 00:31:58.455 --> 00:32:05.120 align:middle line:90% 00:32:05.120 --> 00:32:06.740 align:middle line:84% Well, let me prove the next theorem. 00:32:06.740 --> 00:32:09.860 align:middle line:90% And then, I'll make the comment. 00:32:09.860 --> 00:32:13.900 align:middle line:84% OK, so we have these properties of the Fejer kernel. 00:32:13.900 --> 00:32:16.000 align:middle line:84% And now, what we're going to do is 00:32:16.000 --> 00:32:19.780 align:middle line:84% on our way to proving that we have convergence of the Cesaro 00:32:19.780 --> 00:32:22.935 align:middle line:84% means to a function in L2, we're first 00:32:22.935 --> 00:32:24.560 align:middle line:84% going to do it for continuous function. 00:32:24.560 --> 00:32:28.750 align:middle line:84% So you proved in the assignments that in L2 minus pi to pi, 00:32:28.750 --> 00:32:31.750 align:middle line:84% the continuous functions vanishing at the two endpoints 00:32:31.750 --> 00:32:38.750 align:middle line:84% are dense in the space big L2, OK? 00:32:38.750 --> 00:32:41.600 align:middle line:84% Now, if a function's continuous and equals 0 00:32:41.600 --> 00:32:45.950 align:middle line:84% at both of the endpoints, it's 2 pi periodic in the sense 00:32:45.950 --> 00:32:51.180 align:middle line:84% that it has the same value at both endpoints. 00:32:51.180 --> 00:32:57.150 align:middle line:84% And therefore, the subspace of continuous functions that are 00:32:57.150 --> 00:33:01.020 align:middle line:90% 2 pi periodic is dense in L2. 00:33:01.020 --> 00:33:03.900 align:middle line:84% So if we're going to be able to show that the Cesaro means 00:33:03.900 --> 00:33:08.490 align:middle line:84% converge to a function in L2 for arbitrary L2 function, 00:33:08.490 --> 00:33:10.920 align:middle line:84% maybe it makes sense to try and do it 00:33:10.920 --> 00:33:12.930 align:middle line:90% first for continuous functions. 00:33:12.930 --> 00:33:14.880 align:middle line:84% And it's there that this argument 00:33:14.880 --> 00:33:23.800 align:middle line:84% that I just-- this heuristic argument I gave here 00:33:23.800 --> 00:33:28.954 align:middle line:90% will be more math-like. 00:33:28.954 --> 00:33:31.100 align:middle line:84% OK, so we have a following theorem 00:33:31.100 --> 00:33:41.860 align:middle line:84% due to Fejer, which is the following. 00:33:41.860 --> 00:34:02.320 align:middle line:84% If f is continuous and 2 pi periodic, meaning f of pi 00:34:02.320 --> 00:34:10.679 align:middle line:84% is equal to f minus pi, then not only 00:34:10.679 --> 00:34:13.650 align:middle line:84% do we have the Cesaro means converging to f in L2, 00:34:13.650 --> 00:34:17.550 align:middle line:84% we actually have it in the best sense that you could 00:34:17.550 --> 00:34:18.969 align:middle line:90% for a continuous function. 00:34:18.969 --> 00:34:32.691 align:middle line:84% Then, sigma n of f converges to f uniformly in minus pi to pi, 00:34:32.691 --> 00:34:35.080 align:middle line:90% all right? 00:34:35.080 --> 00:34:40.060 align:middle line:84% So before, we were looking at Fourier series in L2. 00:34:40.060 --> 00:34:43.420 align:middle line:84% So convergence in L2 was the way one 00:34:43.420 --> 00:34:46.690 align:middle line:84% makes sense of infinite series or something 00:34:46.690 --> 00:34:49.270 align:middle line:84% converging to something else, all right? 00:34:49.270 --> 00:34:50.980 align:middle line:84% If we're looking at continuous functions, 00:34:50.980 --> 00:34:53.980 align:middle line:84% then we already a different norm there if we 00:34:53.980 --> 00:34:56.380 align:middle line:84% want to just consider a complete space containing 00:34:56.380 --> 00:34:57.400 align:middle line:90% continuous functions. 00:34:57.400 --> 00:35:00.290 align:middle line:84% We have the uniform norm, or the infinity norm. 00:35:00.290 --> 00:35:06.610 align:middle line:84% And so what this says is that even in this smaller space 00:35:06.610 --> 00:35:10.360 align:middle line:84% and in this stronger norm, we have convergence of the Cesaro 00:35:10.360 --> 00:35:13.510 align:middle line:90% means to the function f. 00:35:13.510 --> 00:35:17.020 align:middle line:84% But again, this doesn't imply that the Fourier series 00:35:17.020 --> 00:35:19.090 align:middle line:90% converges to f uniformly. 00:35:19.090 --> 00:35:24.857 align:middle line:84% Like I said, one can, in fact, use this oscillatory behavior 00:35:24.857 --> 00:35:26.440 align:middle line:84% of the Dirichlet kernel to prove there 00:35:26.440 --> 00:35:31.240 align:middle line:84% exist continuous functions whose Fourier series 00:35:31.240 --> 00:35:33.843 align:middle line:90% diverges at a point. 00:35:33.843 --> 00:35:35.260 align:middle line:84% And therefore, it doesn't converge 00:35:35.260 --> 00:35:37.390 align:middle line:90% uniformly to the function. 00:35:37.390 --> 00:35:41.020 align:middle line:84% But this is true for the Cesaro means 00:35:41.020 --> 00:35:44.140 align:middle line:84% because of these properties of the Fejer kernel, 00:35:44.140 --> 00:35:50.730 align:middle line:84% because it has this shape where it's non-negative. 00:35:50.730 --> 00:35:53.340 align:middle line:90% It's peaking near the origin. 00:35:53.340 --> 00:35:56.850 align:middle line:84% And it has total mass 1, and total integral 1. 00:35:56.850 --> 00:35:58.980 align:middle line:84% In some sense, you should think of, 00:35:58.980 --> 00:36:04.350 align:middle line:84% as n goes to infinity, sigma n is looking more 00:36:04.350 --> 00:36:10.980 align:middle line:84% and more like the Dirac delta function at 0, 00:36:10.980 --> 00:36:13.623 align:middle line:84% which maybe you encountered in physics. 00:36:13.623 --> 00:36:15.790 align:middle line:84% If that doesn't mean anything, don't worry about it. 00:36:15.790 --> 00:36:20.400 align:middle line:84% Just skip to the next part of the talk, which 00:36:20.400 --> 00:36:22.890 align:middle line:84% is supposed to have this magical property that it's 00:36:22.890 --> 00:36:27.750 align:middle line:84% 0 away from 0, which these are looking like, as integral 1. 00:36:27.750 --> 00:36:29.850 align:middle line:84% And when you integrate it against a function, 00:36:29.850 --> 00:36:34.110 align:middle line:84% you get f evaluated at the origin, which is like what 00:36:34.110 --> 00:36:36.190 align:middle line:90% we're saying here, OK? 00:36:36.190 --> 00:36:44.280 align:middle line:84% So again, that's some more heuristics. 00:36:44.280 --> 00:36:51.090 align:middle line:84% But linear operators depending on a parameter that 00:36:51.090 --> 00:36:55.320 align:middle line:84% appear like this, where it's a function of this form times 00:36:55.320 --> 00:36:59.190 align:middle line:84% f of t integrated dt, pop up all the time in harmonic analysis, 00:36:59.190 --> 00:36:59.880 align:middle line:90% OK? 00:36:59.880 --> 00:37:03.640 align:middle line:84% And having these properties, in fact, 00:37:03.640 --> 00:37:06.873 align:middle line:84% pops up also in harmonic analysis, OK? 00:37:06.873 --> 00:37:10.050 align:middle line:90% 00:37:10.050 --> 00:37:13.290 align:middle line:84% Harmonic analysis being a fancy name 00:37:13.290 --> 00:37:20.010 align:middle line:84% for Fourier analysis and other stuff. 00:37:20.010 --> 00:37:22.935 align:middle line:90% So let's prove this. 00:37:22.935 --> 00:37:25.750 align:middle line:90% 00:37:25.750 --> 00:37:29.580 align:middle line:84% So the first thing that I want to do is-- 00:37:29.580 --> 00:37:32.220 align:middle line:84% so f is a continuous function on minus pi to pi. 00:37:32.220 --> 00:37:33.910 align:middle line:90% That's 2 pi periodic. 00:37:33.910 --> 00:37:39.030 align:middle line:84% So I can extend f to all of R by periodicity. 00:37:39.030 --> 00:38:09.340 align:middle line:84% In other words, so we extend to all of R, meaning I have-- 00:38:09.340 --> 00:38:12.910 align:middle line:90% so there's pi minus pi pi. 00:38:12.910 --> 00:38:15.775 align:middle line:90% Here's a 3 pi. 00:38:15.775 --> 00:38:18.320 align:middle line:90% 00:38:18.320 --> 00:38:19.700 align:middle line:90% Here's minus 3 pi. 00:38:19.700 --> 00:38:22.700 align:middle line:84% So supposedly, I have this continuous function, 00:38:22.700 --> 00:38:24.200 align:middle line:90% which is 2 pi periodic. 00:38:24.200 --> 00:38:26.330 align:middle line:84% Now, I take that continuous function 00:38:26.330 --> 00:38:36.470 align:middle line:84% and just extend it by how it is here and so on, OK? 00:38:36.470 --> 00:38:39.170 align:middle line:84% I'm not saying I extend it by 0 outside. 00:38:39.170 --> 00:38:42.176 align:middle line:84% I'm saying I extend it periodically, OK? 00:38:42.176 --> 00:38:47.660 align:middle line:90% 00:38:47.660 --> 00:38:49.943 align:middle line:84% OK, now, I can write down a formula 00:38:49.943 --> 00:38:51.110 align:middle line:90% for exactly how you do that. 00:38:51.110 --> 00:38:53.360 align:middle line:90% But just trust me. 00:38:53.360 --> 00:38:55.340 align:middle line:90% You can do that. 00:38:55.340 --> 00:38:58.550 align:middle line:84% And also the following simple properties, then-- 00:38:58.550 --> 00:39:01.920 align:middle line:90% 00:39:01.920 --> 00:39:06.810 align:middle line:84% f, now referring to it as a function defined 00:39:06.810 --> 00:39:09.900 align:middle line:84% on all of R that's 2 pi periodic, 00:39:09.900 --> 00:39:20.340 align:middle line:84% this is also continuous, is 2 pi periodic, which 00:39:20.340 --> 00:39:37.250 align:middle line:84% implies that f is uniformly continuous and bounded, i.e. 00:39:37.250 --> 00:39:47.950 align:middle line:84% If I look at the infinity norm of f 00:39:47.950 --> 00:39:50.950 align:middle line:84% first off, because by periodicity, this is just 00:39:50.950 --> 00:40:01.090 align:middle line:84% equal to sup xn minus pi to pi, and because f is continuous, 00:40:01.090 --> 00:40:03.661 align:middle line:90% this thing is finite, OK? 00:40:03.661 --> 00:40:07.480 align:middle line:90% 00:40:07.480 --> 00:40:08.020 align:middle line:90% All right. 00:40:08.020 --> 00:40:12.790 align:middle line:84% Now, it's not difficult to believe that f is-- 00:40:12.790 --> 00:40:16.570 align:middle line:84% if I extend it by periodicity, it's going to be continuous. 00:40:16.570 --> 00:40:18.970 align:middle line:84% But using that and the fact that it's 2 pi periodic, 00:40:18.970 --> 00:40:20.470 align:middle line:84% you can then also conclude that it's 00:40:20.470 --> 00:40:25.870 align:middle line:90% uniformly continuous, meaning-- 00:40:25.870 --> 00:40:28.600 align:middle line:84% let's just quickly review what uniformly continuous means. 00:40:28.600 --> 00:40:31.490 align:middle line:84% This means for all epsilon positive, 00:40:31.490 --> 00:40:34.150 align:middle line:84% there exists a delta positive such 00:40:34.150 --> 00:40:45.200 align:middle line:84% that if y minus z is less than delta, then f of y minus f of z 00:40:45.200 --> 00:40:48.470 align:middle line:84% is less than epsilon, meaning I can choose a delta independent 00:40:48.470 --> 00:40:50.780 align:middle line:90% of y and of the point, right? 00:40:50.780 --> 00:40:53.090 align:middle line:84% Continuous at a point means I fix x. 00:40:53.090 --> 00:40:55.220 align:middle line:84% Then, for all epsilon, there exists a delta. 00:40:55.220 --> 00:40:58.490 align:middle line:84% Uniformly continuous means the delta 00:40:58.490 --> 00:41:05.870 align:middle line:84% doesn't depend on x, the point that I'm looking at. 00:41:05.870 --> 00:41:09.190 align:middle line:90% 00:41:09.190 --> 00:41:25.650 align:middle line:84% All right, so we have basic observation 00:41:25.650 --> 00:41:26.950 align:middle line:90% that we're going to make there. 00:41:26.950 --> 00:41:31.020 align:middle line:84% And maybe I'll just leave this up for now. 00:41:31.020 --> 00:41:35.390 align:middle line:90% 00:41:35.390 --> 00:41:39.620 align:middle line:84% So we want to prove the sigma n's converge to f 00:41:39.620 --> 00:41:42.620 align:middle line:90% uniformly on minus pi to pi. 00:41:42.620 --> 00:41:45.770 align:middle line:84% So that means we should be able to find, for every epsilon, 00:41:45.770 --> 00:41:50.900 align:middle line:84% a capital M such that for all n bigger than or equal to M 00:41:50.900 --> 00:41:55.340 align:middle line:84% sigma int and for all x in minus pi to pi sigma n of f minus f 00:41:55.340 --> 00:41:59.580 align:middle line:84% is less than epsilon in absolute value. 00:41:59.580 --> 00:42:05.080 align:middle line:84% All right, so let epsilon be positive. 00:42:05.080 --> 00:42:09.340 align:middle line:90% 00:42:09.340 --> 00:42:16.090 align:middle line:84% Since f is uniformly continuous, as I stated-- 00:42:16.090 --> 00:42:19.660 align:middle line:84% recalling the definition-- that implies that there exists 00:42:19.660 --> 00:42:28.460 align:middle line:84% a delta positive such that if y minus z is less than delta, 00:42:28.460 --> 00:42:34.690 align:middle line:84% then f of y minus f of z is less than-- 00:42:34.690 --> 00:42:38.120 align:middle line:84% and let me get this right so it comes out pretty in the end-- 00:42:38.120 --> 00:42:42.977 align:middle line:90% is less than epsilon over 2, OK? 00:42:42.977 --> 00:42:44.560 align:middle line:84% So now, what we're going to go through 00:42:44.560 --> 00:42:47.020 align:middle line:84% is make that argument which I just erased 00:42:47.020 --> 00:42:50.620 align:middle line:90% actually precise, all right? 00:42:50.620 --> 00:42:55.900 align:middle line:84% So here, we're saying if f is very close to-- 00:42:55.900 --> 00:42:58.870 align:middle line:84% if any two points are sufficiently close, 00:42:58.870 --> 00:43:00.745 align:middle line:90% f is going to be close in value. 00:43:00.745 --> 00:43:07.820 align:middle line:90% 00:43:07.820 --> 00:43:15.230 align:middle line:84% OK, now choose M natural number so 00:43:15.230 --> 00:43:23.220 align:middle line:84% that for all n bigger than or equal to M, the quantity 00:43:23.220 --> 00:43:30.840 align:middle line:84% twice times the L infinity norm over n plus 1 times sine 00:43:30.840 --> 00:43:36.120 align:middle line:84% squared delta over 2 is less than epsilon over 2, OK? 00:43:36.120 --> 00:43:39.060 align:middle line:84% So n plus 1-- that's the thing that's changing. 00:43:39.060 --> 00:43:41.820 align:middle line:84% So I have these fixed numbers here now. 00:43:41.820 --> 00:43:43.050 align:middle line:90% I've fixed delta. 00:43:43.050 --> 00:43:45.120 align:middle line:90% I have the L infinity norm of f. 00:43:45.120 --> 00:43:46.470 align:middle line:90% So I have this number here. 00:43:46.470 --> 00:43:50.130 align:middle line:84% And I'm just saying, choose a capital M so that for all n 00:43:50.130 --> 00:43:52.560 align:middle line:84% bigger than or equal to M, this number of times-- 00:43:52.560 --> 00:43:53.850 align:middle line:90% and I'll even put it here-- 00:43:53.850 --> 00:43:57.660 align:middle line:84% times 1 over n plus 1 is small, is less than epsilon over 2. 00:43:57.660 --> 00:44:01.110 align:middle line:84% And I can do that because this, as capital N goes to infinity, 00:44:01.110 --> 00:44:06.190 align:middle line:90% converges to 0, right? 00:44:06.190 --> 00:44:06.690 align:middle line:90% OK. 00:44:06.690 --> 00:44:15.240 align:middle line:84% Now, since f and k sub n, the Dirichlet kernel, 00:44:15.240 --> 00:44:32.640 align:middle line:84% are 2 pi periodic, I can write the Cesaro mean, 00:44:32.640 --> 00:44:44.380 align:middle line:84% which is given by minus pi to pi kn of x minus t f of t dt. 00:44:44.380 --> 00:44:46.010 align:middle line:84% I can make a change of variables, 00:44:46.010 --> 00:44:50.620 align:middle line:90% set tau equal to x minus t. 00:44:50.620 --> 00:44:52.195 align:middle line:84% And then, this will be equal to-- 00:44:52.195 --> 00:44:59.670 align:middle line:90% 00:44:59.670 --> 00:45:02.730 align:middle line:90% what is it going to be equal to? 00:45:02.730 --> 00:45:17.930 align:middle line:84% x minus pi x plus pi kn of tau f of x minus tau d tau, OK? 00:45:17.930 --> 00:45:19.550 align:middle line:84% All of this change of variable stuff 00:45:19.550 --> 00:45:22.790 align:middle line:84% is fine, because I'm dealing with continuous functions. 00:45:22.790 --> 00:45:24.600 align:middle line:84% I'm integrating continuous functions. 00:45:24.600 --> 00:45:26.030 align:middle line:90% So that's the Riemann integral. 00:45:26.030 --> 00:45:29.100 align:middle line:84% We have a change of variables for the Riemann integral. 00:45:29.100 --> 00:45:31.114 align:middle line:90% So that's completely fine. 00:45:31.114 --> 00:45:37.890 align:middle line:84% OK, now this is the product of 2 pi periodic functions. 00:45:37.890 --> 00:45:41.610 align:middle line:84% And if I take the integral of that quantity of a 2 pi 00:45:41.610 --> 00:45:45.540 align:middle line:84% periodic function, the integral of that 00:45:45.540 --> 00:45:50.740 align:middle line:84% is equal to the integral over any interval of length 2 pi, 00:45:50.740 --> 00:45:51.240 align:middle line:90% all right? 00:45:51.240 --> 00:45:56.010 align:middle line:84% So we're integrating over an interval of length 2 pi, right? 00:45:56.010 --> 00:45:58.140 align:middle line:84% We're going from x minus pi to x plus pi. 00:45:58.140 --> 00:46:01.470 align:middle line:84% That is equal to the integral of the same quantity 00:46:01.470 --> 00:46:03.420 align:middle line:84% over any interval of length 2 pi. 00:46:03.420 --> 00:46:07.545 align:middle line:84% So it's also equal to the integral over minus pi to pi. 00:46:07.545 --> 00:46:14.450 align:middle line:90% 00:46:14.450 --> 00:46:15.140 align:middle line:90% OK? 00:46:15.140 --> 00:46:19.160 align:middle line:84% So all I'm saying is I can change variables and move 00:46:19.160 --> 00:46:20.780 align:middle line:90% the x minus t. 00:46:20.780 --> 00:46:25.970 align:middle line:84% And let me even go back to t instead of tau here. 00:46:25.970 --> 00:46:31.395 align:middle line:84% Because of periodicity, I can switch this x minus t over here 00:46:31.395 --> 00:46:31.895 align:middle line:90% to f. 00:46:31.895 --> 00:46:36.518 align:middle line:90% 00:46:36.518 --> 00:46:39.060 align:middle line:84% All right, now we're going to start seeing some magic happen. 00:46:39.060 --> 00:46:44.030 align:middle line:90% 00:46:44.030 --> 00:46:48.430 align:middle line:84% And this is where that heuristic argument that I gave earlier 00:46:48.430 --> 00:46:52.690 align:middle line:90% actually starts to make sense. 00:46:52.690 --> 00:47:06.480 align:middle line:90% 00:47:06.480 --> 00:47:13.910 align:middle line:84% So then, I have that for all n bigger than 00:47:13.910 --> 00:47:17.720 align:middle line:84% or equal to M-- so I have that condition 00:47:17.720 --> 00:47:20.840 align:middle line:84% that quantity was less than epsilon over 2. 00:47:20.840 --> 00:47:24.860 align:middle line:84% And for all x and minus pi to pi, 00:47:24.860 --> 00:47:33.690 align:middle line:84% I have that sigma n f of x minus f of x-- 00:47:33.690 --> 00:47:37.140 align:middle line:84% so this is equal to minus pi to pi. 00:47:37.140 --> 00:47:41.180 align:middle line:90% 00:47:41.180 --> 00:47:43.070 align:middle line:84% And again, I'm going to write this now 00:47:43.070 --> 00:47:51.230 align:middle line:84% as kn of t f of x minus t dt minus-- 00:47:51.230 --> 00:47:54.450 align:middle line:90% now, here's the trick. 00:47:54.450 --> 00:47:56.660 align:middle line:90% The Fejer kernel has integral 1. 00:47:56.660 --> 00:48:00.530 align:middle line:84% So I can actually write f of x as minus pi 00:48:00.530 --> 00:48:07.340 align:middle line:84% to pi integral kn of t times f of x dt. 00:48:07.340 --> 00:48:09.280 align:middle line:90% I'm integrating dt, right? 00:48:09.280 --> 00:48:10.680 align:middle line:90% Then this just pops out. 00:48:10.680 --> 00:48:13.220 align:middle line:84% I get f of x times the integral of the Fejer kernel, 00:48:13.220 --> 00:48:16.240 align:middle line:90% which is 1, OK? 00:48:16.240 --> 00:48:19.180 align:middle line:90% And this equals minus pi pi. 00:48:19.180 --> 00:48:21.250 align:middle line:90% So just combining things-- 00:48:21.250 --> 00:48:26.960 align:middle line:84% kn of t f of x minus t minus f of x, 00:48:26.960 --> 00:48:29.240 align:middle line:84% which is good, because we have a continuous function. 00:48:29.240 --> 00:48:31.450 align:middle line:84% And now, we have something inside that 00:48:31.450 --> 00:48:35.920 align:middle line:84% looks like I'm subtracting f of some argument minus f 00:48:35.920 --> 00:48:40.330 align:middle line:84% of the argument minus something, OK? 00:48:40.330 --> 00:48:42.730 align:middle line:84% Now, I'm going to split this integral into two parts, 00:48:42.730 --> 00:48:44.260 align:middle line:84% and then use the triangle inequality 00:48:44.260 --> 00:48:46.240 align:middle line:84% and bring the triangle inequality inside. 00:48:46.240 --> 00:48:48.240 align:middle line:84% In fact, I'm going to go ahead and do that here. 00:48:48.240 --> 00:48:51.640 align:middle line:84% This is less than or equal to if I combine terms like I did 00:48:51.640 --> 00:48:53.950 align:middle line:84% and then bring the absolute value inside. 00:48:53.950 --> 00:48:58.460 align:middle line:90% 00:48:58.460 --> 00:49:00.190 align:middle line:90% OK? 00:49:00.190 --> 00:49:04.430 align:middle line:84% And now, I'm going to split this integral up into two parts. 00:49:04.430 --> 00:49:11.890 align:middle line:84% This is equal to the integral over t less than delta. 00:49:11.890 --> 00:49:14.440 align:middle line:84% And because kn is non-negative, this 00:49:14.440 --> 00:49:18.790 align:middle line:84% is just kn of t f of x minus t minus f 00:49:18.790 --> 00:49:24.180 align:middle line:90% of x dt plus the other term. 00:49:24.180 --> 00:49:31.350 align:middle line:90% 00:49:31.350 --> 00:49:31.850 align:middle line:90% OK? 00:49:31.850 --> 00:49:36.500 align:middle line:90% 00:49:36.500 --> 00:49:44.660 align:middle line:84% kn of t f of x minus t minus f of t dt, all right? 00:49:44.660 --> 00:49:46.670 align:middle line:90% Now, what do we know? 00:49:46.670 --> 00:49:49.790 align:middle line:84% If the absolute value of t is less than delta, then x 00:49:49.790 --> 00:49:53.000 align:middle line:84% minus t minus x is equal to minus t, 00:49:53.000 --> 00:49:56.340 align:middle line:84% which is an absolute value less than delta. 00:49:56.340 --> 00:50:02.870 align:middle line:84% So note that x minus t minus x equals t is less than delta 00:50:02.870 --> 00:50:05.320 align:middle line:90% here, right? 00:50:05.320 --> 00:50:07.960 align:middle line:84% And therefore, this quantity here 00:50:07.960 --> 00:50:10.810 align:middle line:84% is less than epsilon over 2 by how we chose delta. 00:50:10.810 --> 00:50:16.630 align:middle line:84% So this is less than epsilon over 2 times the integral 00:50:16.630 --> 00:50:19.100 align:middle line:90% over this region of kn of t. 00:50:19.100 --> 00:50:19.780 align:middle line:90% OK? 00:50:19.780 --> 00:50:23.820 align:middle line:84% But I can make this region larger and just go back to-- 00:50:23.820 --> 00:50:26.625 align:middle line:84% so let me just leave it here as it is. 00:50:26.625 --> 00:50:32.580 align:middle line:90% 00:50:32.580 --> 00:50:36.300 align:middle line:84% Plus now what do I do with this piece? 00:50:36.300 --> 00:50:38.660 align:middle line:90% I have this. 00:50:38.660 --> 00:50:42.445 align:middle line:84% I bound by twice the L infinity norm of f. 00:50:42.445 --> 00:50:44.570 align:middle line:84% The absolute value of this is less than or equal to 00:50:44.570 --> 00:50:46.610 align:middle line:84% by the triangle inequality, the sum 00:50:46.610 --> 00:50:48.560 align:middle line:84% of the absolute values, which is less 00:50:48.560 --> 00:50:52.650 align:middle line:84% than or equal to the sup of this plus the sup of that and x, 00:50:52.650 --> 00:50:55.340 align:middle line:84% which is equal to twice the infinity norm. 00:50:55.340 --> 00:51:01.760 align:middle line:84% So I get 2 times the infinity norm of f popping out 00:51:01.760 --> 00:51:04.880 align:middle line:90% from this term, and kn of t-- 00:51:04.880 --> 00:51:08.990 align:middle line:84% oh, I'm away from t less than delta. 00:51:08.990 --> 00:51:11.720 align:middle line:84% And this is where I use that third property 00:51:11.720 --> 00:51:19.560 align:middle line:84% that I have from before, that it's less than 1 over 2 pi. 00:51:19.560 --> 00:51:20.960 align:middle line:90% So let me leave this here. 00:51:20.960 --> 00:51:28.730 align:middle line:90% 00:51:28.730 --> 00:51:37.730 align:middle line:84% Sum 1 over 2 pi n plus 1, sine squared delta over 2 dt, OK? 00:51:37.730 --> 00:51:41.210 align:middle line:84% And now, this, I can say, is less than 00:51:41.210 --> 00:51:44.150 align:middle line:84% or equal to the whole integral over minus pi 00:51:44.150 --> 00:51:45.500 align:middle line:90% to pi, which is equal to 1. 00:51:45.500 --> 00:51:51.020 align:middle line:90% 00:51:51.020 --> 00:51:55.820 align:middle line:84% Plus again, making this an integral 00:51:55.820 --> 00:51:59.720 align:middle line:84% over the entire region, I get 2 pi times-- 00:51:59.720 --> 00:52:01.760 align:middle line:90% or divided by 2 pi gives me 1. 00:52:01.760 --> 00:52:11.950 align:middle line:84% So I get twice infinity over n plus 1 sine squared delta 00:52:11.950 --> 00:52:12.610 align:middle line:90% over 2. 00:52:12.610 --> 00:52:16.870 align:middle line:84% And we chose n plus 1 so that this second quantity here 00:52:16.870 --> 00:52:18.265 align:middle line:90% is less than epsilon over 2. 00:52:18.265 --> 00:52:22.480 align:middle line:90% 00:52:22.480 --> 00:52:25.230 align:middle line:90% OK? 00:52:25.230 --> 00:52:31.200 align:middle line:84% And therefore, uniformly, we prove that for all capital 00:52:31.200 --> 00:52:35.520 align:middle line:84% N bigger than or equal to M for all x in minus pi to pi, 00:52:35.520 --> 00:52:40.830 align:middle line:84% the difference in sigma nf in f is less than epsilon, proving 00:52:40.830 --> 00:52:43.570 align:middle line:90% uniform convergence, OK? 00:52:43.570 --> 00:52:47.470 align:middle line:90% 00:52:47.470 --> 00:52:54.610 align:middle line:84% So here's the remark I was going to make, 00:52:54.610 --> 00:53:28.120 align:middle line:84% is that the same proof can be modified if instead of kn 00:53:28.120 --> 00:53:29.872 align:middle line:84% of x being bigger than or equal to 0-- 00:53:29.872 --> 00:53:31.705 align:middle line:84% let me make sure I'm saying the right thing. 00:53:31.705 --> 00:53:35.950 align:middle line:90% 00:53:35.950 --> 00:53:39.570 align:middle line:84% So if instead of this property, which we had for the Fejer 00:53:39.570 --> 00:53:54.460 align:middle line:84% kernel, we have that sup over n of the integral from minus pi 00:53:54.460 --> 00:54:06.410 align:middle line:84% to pi of kn of x is finite, meaning if I have a function 00:54:06.410 --> 00:54:09.500 align:middle line:84% or if I have a sequence of functions, 00:54:09.500 --> 00:54:13.250 align:middle line:84% kn's, and I have the corresponding operators that 00:54:13.250 --> 00:54:16.070 align:middle line:84% look like that-- maybe they're not associated to any questions 00:54:16.070 --> 00:54:19.510 align:middle line:84% about Fourier analysis, but I'm just saying-- 00:54:19.510 --> 00:54:23.500 align:middle line:84% and it satisfies the three properties I had before 00:54:23.500 --> 00:54:25.400 align:middle line:84% with the exception of being non-negative, 00:54:25.400 --> 00:54:28.160 align:middle line:84% but instead of that, it satisfies this property, 00:54:28.160 --> 00:54:30.370 align:middle line:84% then I can do redo the same proof 00:54:30.370 --> 00:54:36.420 align:middle line:84% and show that those things converge to f uniformly, OK? 00:54:36.420 --> 00:54:38.020 align:middle line:90% Why am I saying that? 00:54:38.020 --> 00:54:43.060 align:middle line:84% Because maybe you would like to then try 00:54:43.060 --> 00:54:49.210 align:middle line:84% your hand at replacing kn with dn, the Dirichlet kernel, OK? 00:54:49.210 --> 00:54:52.450 align:middle line:84% The Dirichlet kernel satisfies all of the other properties 00:54:52.450 --> 00:54:53.410 align:middle line:90% we had up there. 00:54:53.410 --> 00:54:55.660 align:middle line:90% The integral is 1. 00:54:55.660 --> 00:55:01.180 align:middle line:84% In absolute value, it decays away from x is less than delta. 00:55:01.180 --> 00:55:07.250 align:middle line:84% And it's even in 2 pi periodic, OK? 00:55:07.250 --> 00:55:08.710 align:middle line:90% But it doesn't satisfy this. 00:55:08.710 --> 00:55:11.870 align:middle line:90% 00:55:11.870 --> 00:55:20.030 align:middle line:84% And if I look at minus pi to pi of the Dirichlet kernel, 00:55:20.030 --> 00:55:22.010 align:middle line:84% what one can prove is that this is 00:55:22.010 --> 00:55:29.607 align:middle line:84% something like log n for large enough n, OK? 00:55:29.607 --> 00:55:32.946 align:middle line:90% 00:55:32.946 --> 00:55:34.350 align:middle line:90% All right? 00:55:34.350 --> 00:55:36.600 align:middle line:84% So that was just a tiny remark I wanted 00:55:36.600 --> 00:55:39.390 align:middle line:84% to say on why, if you thought about maybe redoing 00:55:39.390 --> 00:55:42.090 align:middle line:84% this proof using the Dirichlet kernel, which satisfies almost 00:55:42.090 --> 00:55:44.400 align:middle line:84% all the same properties with the exception of being 00:55:44.400 --> 00:55:47.985 align:middle line:84% non-negative, you could, if the Dirichlet kernel 00:55:47.985 --> 00:55:49.410 align:middle line:90% had satisfied this bound. 00:55:49.410 --> 00:55:50.040 align:middle line:90% But it doesn't. 00:55:50.040 --> 00:55:51.510 align:middle line:90% It satisfies this bound. 00:55:51.510 --> 00:55:52.350 align:middle line:90% It's like log n. 00:55:52.350 --> 00:55:53.730 align:middle line:84% And therefore, if I take the sup, 00:55:53.730 --> 00:55:58.190 align:middle line:84% I don't get something finite, OK? 00:55:58.190 --> 00:56:02.810 align:middle line:84% All right, so we've proven that the Cesaro 00:56:02.810 --> 00:56:05.660 align:middle line:84% means of a continuous function convert uniformly 00:56:05.660 --> 00:56:07.265 align:middle line:90% to a continuous function. 00:56:07.265 --> 00:56:17.393 align:middle line:90% 00:56:17.393 --> 00:56:18.810 align:middle line:84% So we're almost to the point where 00:56:18.810 --> 00:56:22.810 align:middle line:84% we can say that the Cesaro means of an L2 00:56:22.810 --> 00:56:25.470 align:middle line:84% to function converge to an L2 function 00:56:25.470 --> 00:56:33.270 align:middle line:84% and conclude that the subset of exponentials divided 00:56:33.270 --> 00:56:38.490 align:middle line:84% by square root of 2 pi form a maximal orthonormal subset 00:56:38.490 --> 00:56:41.920 align:middle line:84% of L2, and therefore is in orthonormal basis 00:56:41.920 --> 00:56:46.650 align:middle line:84% so that the partial Fourier sums converge back 00:56:46.650 --> 00:56:49.910 align:middle line:90% to the function in L2. 00:56:49.910 --> 00:56:53.935 align:middle line:84% We just need one more bit of information. 00:56:53.935 --> 00:56:55.310 align:middle line:84% So we have the following theorem. 00:56:55.310 --> 00:56:58.580 align:middle line:90% 00:56:58.580 --> 00:57:05.660 align:middle line:84% For all f in L2 of minus pi to pi, 00:57:05.660 --> 00:57:09.920 align:middle line:84% if I look at sigma n of f-- so first off, 00:57:09.920 --> 00:57:14.520 align:middle line:84% this is just a finite linear combination of exponentials, 00:57:14.520 --> 00:57:15.020 align:middle line:90% right? 00:57:15.020 --> 00:57:17.030 align:middle line:90% So this is clearly an L2. 00:57:17.030 --> 00:57:19.110 align:middle line:90% It's a continuous function. 00:57:19.110 --> 00:57:23.120 align:middle line:84% But if I take the L2 norm of that, it could depend on n. 00:57:23.120 --> 00:57:27.800 align:middle line:84% But in fact, it's less than or equal to the L2 norm of f. 00:57:27.800 --> 00:57:31.000 align:middle line:90% 00:57:31.000 --> 00:57:31.500 align:middle line:90% OK? 00:57:31.500 --> 00:57:35.790 align:middle line:84% So how we'll prove this is we'll first-- 00:57:35.790 --> 00:57:41.010 align:middle line:84% and this bound is what allows us to go from the 2 pi 00:57:41.010 --> 00:57:44.670 align:middle line:84% periodic continuous functions to general L2 functions 00:57:44.670 --> 00:57:48.370 align:middle line:90% by a density argument, OK? 00:57:48.370 --> 00:57:51.370 align:middle line:84% So first, we'll do this for 2 pi periodic continuous functions, 00:57:51.370 --> 00:57:56.000 align:middle line:84% and then by density, conclude it for L2 functions. 00:57:56.000 --> 00:58:07.540 align:middle line:84% So suppose first that f is 2 pi periodic. 00:58:07.540 --> 00:58:14.820 align:middle line:90% 00:58:14.820 --> 00:58:18.378 align:middle line:84% And then of course, extend it to R by periodicity 00:58:18.378 --> 00:58:19.170 align:middle line:90% like we did before. 00:58:19.170 --> 00:58:24.040 align:middle line:90% 00:58:24.040 --> 00:58:30.760 align:middle line:84% Then, as before, we had that the Cesaro mean of f 00:58:30.760 --> 00:58:33.970 align:middle line:84% is equal to the integral from minus pi to pi 00:58:33.970 --> 00:58:39.730 align:middle line:90% of f of x minus t kn of t dt. 00:58:39.730 --> 00:58:50.850 align:middle line:84% And so if I compute the integral sigma n f of x squared dx, 00:58:50.850 --> 00:58:52.190 align:middle line:90% this is equal to-- 00:58:52.190 --> 00:58:55.410 align:middle line:84% so each one of these is equal to an integral over minus pi 00:58:55.410 --> 00:58:55.910 align:middle line:90% to pi. 00:58:55.910 --> 00:58:58.100 align:middle line:84% So I'm going to have three integrals. 00:58:58.100 --> 00:59:05.100 align:middle line:90% 00:59:05.100 --> 00:59:15.940 align:middle line:84% And f of x minus s times the complex conjugate 00:59:15.940 --> 00:59:30.530 align:middle line:84% fx minus t times kn of s and kn of t, And Then ds dt dx, OK? 00:59:30.530 --> 00:59:33.200 align:middle line:90% 00:59:33.200 --> 00:59:40.140 align:middle line:84% Now, all these functions are continuous. 00:59:40.140 --> 00:59:42.770 align:middle line:84% So we have a Fubini's theorem, which 00:59:42.770 --> 00:59:48.240 align:middle line:84% says we can reverse the order of integration however we please. 00:59:48.240 --> 00:59:50.600 align:middle line:84% So I can write this as the integral for minus 00:59:50.600 --> 00:59:57.050 align:middle line:84% pi to pi, minus pi to pi, of now integrating first 00:59:57.050 --> 00:59:58.115 align:middle line:90% with respect to x-- 00:59:58.115 --> 01:00:03.590 align:middle line:90% 01:00:03.590 --> 01:00:05.450 align:middle line:90% kn t. 01:00:05.450 --> 01:00:08.060 align:middle line:84% And now, integrating first with respect to x. 01:00:08.060 --> 01:00:16.310 align:middle line:90% 01:00:16.310 --> 01:00:23.970 align:middle line:90% dx, let's say ds, dt, OK? 01:00:23.970 --> 01:00:29.770 align:middle line:90% Now I do Cauchy-Schwarz on this. 01:00:29.770 --> 01:00:34.320 align:middle line:84% And so this is less than or equal to minus 01:00:34.320 --> 01:00:43.770 align:middle line:84% pi to pi minus pi to pi pi kn of s kn of t times-- 01:00:43.770 --> 01:00:46.110 align:middle line:84% I'm using Cauchy-Schwarz in x now-- so 01:00:46.110 --> 01:00:51.870 align:middle line:84% times the L2 norm of the function minus s. 01:00:51.870 --> 01:00:54.660 align:middle line:84% So I'm taking the L2 norm in this variable-- 01:00:54.660 --> 01:00:57.790 align:middle line:90% 01:00:57.790 --> 01:01:08.200 align:middle line:90% 2 times 2 ds dt, OK? 01:01:08.200 --> 01:01:09.700 align:middle line:84% What I mean by this is I'm taking 01:01:09.700 --> 01:01:13.420 align:middle line:84% the L2 norm of this function depending on s, 01:01:13.420 --> 01:01:17.450 align:middle line:84% But In the first variable-- in this x variable, OK? 01:01:17.450 --> 01:01:19.700 align:middle line:84% So just write it out to see what I mean. 01:01:19.700 --> 01:01:26.070 align:middle line:84% Now, this is the integral of a function over an interval 01:01:26.070 --> 01:01:27.270 align:middle line:90% of length 2 pi. 01:01:27.270 --> 01:01:28.380 align:middle line:90% That's 2 pi periodic. 01:01:28.380 --> 01:01:32.490 align:middle line:84% That's equal to the integral of that function over any 2 pi 01:01:32.490 --> 01:01:36.510 align:middle line:84% periodic interval or any interval of length 2 pi. 01:01:36.510 --> 01:01:40.290 align:middle line:84% So I can, in fact, remove this s and remove this t, 01:01:40.290 --> 01:01:44.620 align:middle line:84% and just pick up the L2 norm of f in both places. 01:01:44.620 --> 01:01:47.670 align:middle line:90% So this is, in fact, equal to-- 01:01:47.670 --> 01:01:51.950 align:middle line:84% and because these two things no longer depend on s and t, 01:01:51.950 --> 01:01:53.700 align:middle line:84% they come all the way out of the integral. 01:01:53.700 --> 01:02:03.810 align:middle line:84% And I get L2 norm squared times minus pi to pi kn of s ds times 01:02:03.810 --> 01:02:09.330 align:middle line:84% the integral from minus pi to pi kn of t dt. 01:02:09.330 --> 01:02:12.210 align:middle line:90% Both of these integrals equal 1. 01:02:12.210 --> 01:02:15.960 align:middle line:90% So I get norm squared. 01:02:15.960 --> 01:02:19.740 align:middle line:84% And I started off with the L2 norm squared, 01:02:19.740 --> 01:02:24.120 align:middle line:84% or the L2 norm squared of the Cesaro mean of f. 01:02:24.120 --> 01:02:34.817 align:middle line:84% So I get this for all 2 pi periodic continuous functions, 01:02:34.817 --> 01:02:36.190 align:middle line:90% OK? 01:02:36.190 --> 01:02:40.360 align:middle line:84% Now, how do we then get the bound for general f? 01:02:40.360 --> 01:02:42.040 align:middle line:90% We use the density argument. 01:02:42.040 --> 01:02:52.380 align:middle line:84% So by what you've done in the assignments, 01:02:52.380 --> 01:02:57.345 align:middle line:84% there exists a sequence of 2 pi-- 01:02:57.345 --> 01:03:00.790 align:middle line:90% 01:03:00.790 --> 01:03:06.950 align:middle line:90% so let me start over real quick. 01:03:06.950 --> 01:03:13.212 align:middle line:84% Now, let's take a general element in L2. 01:03:13.212 --> 01:03:13.920 align:middle line:90% OK, now we start. 01:03:13.920 --> 01:03:20.990 align:middle line:84% By assignments, you know that there 01:03:20.990 --> 01:03:25.670 align:middle line:84% exists a sequence of 2 pi periodic continuous functions 01:03:25.670 --> 01:03:26.930 align:middle line:90% converging to f in L2-- 01:03:26.930 --> 01:03:29.960 align:middle line:90% 01:03:29.960 --> 01:03:40.750 align:middle line:84% fn a of 2 pi periodic continuous functions such 01:03:40.750 --> 01:03:48.465 align:middle line:84% that the fn's converge to f in L2. 01:03:48.465 --> 01:03:51.320 align:middle line:90% 01:03:51.320 --> 01:03:54.980 align:middle line:84% And one can verify simply from the definition of each 01:03:54.980 --> 01:03:59.360 align:middle line:90% of the Cesaro means that then-- 01:03:59.360 --> 01:04:01.880 align:middle line:84% so this is as little n goes to infinity-- 01:04:01.880 --> 01:04:08.180 align:middle line:90% 01:04:08.180 --> 01:04:17.730 align:middle line:84% that then the Cesaro means also converge 01:04:17.730 --> 01:04:19.330 align:middle line:90% as little n goes to infinity. 01:04:19.330 --> 01:04:22.685 align:middle line:90% So capital N here is fixed, OK? 01:04:22.685 --> 01:04:24.060 align:middle line:84% Just using the definition of what 01:04:24.060 --> 01:04:28.878 align:middle line:84% the Cesaro mean is and Cauchy-Schwarz, basically, OK? 01:04:28.878 --> 01:04:33.010 align:middle line:90% 01:04:33.010 --> 01:04:35.830 align:middle line:84% And the fact that fn's converge to f in L2. 01:04:35.830 --> 01:04:42.700 align:middle line:84% Thus we get that the L2 norm of the Cesaro mean 01:04:42.700 --> 01:04:46.300 align:middle line:84% is equal to the limit, as n goes to infinity, of-- 01:04:46.300 --> 01:04:49.360 align:middle line:90% 01:04:49.360 --> 01:04:51.740 align:middle line:90% so this is little n-- 01:04:51.740 --> 01:04:55.040 align:middle line:84% of the L2 norm of the Cesaro means of these continuous 2 pi 01:04:55.040 --> 01:04:58.860 align:middle line:84% periodic functions, which as we've proven already-- 01:04:58.860 --> 01:05:05.000 align:middle line:84% these are all less than or equal to the L2 norm of fn, 01:05:05.000 --> 01:05:06.500 align:middle line:90% because they're 2 pi periodic. 01:05:06.500 --> 01:05:10.280 align:middle line:84% And again, because f is converging to fn, 01:05:10.280 --> 01:05:11.210 align:middle line:90% the norms converge. 01:05:11.210 --> 01:05:15.170 align:middle line:90% 01:05:15.170 --> 01:05:21.820 align:middle line:84% And I get the result I wanted for general L2 functions. 01:05:21.820 --> 01:05:22.320 align:middle line:90% OK? 01:05:22.320 --> 01:05:23.445 align:middle line:90% So now, we're almost there. 01:05:23.445 --> 01:05:29.230 align:middle line:90% 01:05:29.230 --> 01:05:31.020 align:middle line:90% What we have is this bound. 01:05:31.020 --> 01:05:37.240 align:middle line:84% And we have that the Cesaro remains converge to-- 01:05:37.240 --> 01:05:40.690 align:middle line:84% so if I take the Cesaro means of a continuous function, 01:05:40.690 --> 01:05:42.670 align:middle line:84% those converge to the continuous function 01:05:42.670 --> 01:05:44.740 align:middle line:90% uniformly on the interval. 01:05:44.740 --> 01:05:47.890 align:middle line:84% We're going to use that, this bound, and the density, again, 01:05:47.890 --> 01:05:53.560 align:middle line:84% of the 2 pi periodic continuous functions in L2 to conclude 01:05:53.560 --> 01:05:54.565 align:middle line:90% the following theorem. 01:05:54.565 --> 01:05:59.090 align:middle line:90% 01:05:59.090 --> 01:06:12.290 align:middle line:84% For all f in L2, the Cesaro means 01:06:12.290 --> 01:06:19.020 align:middle line:84% converge to f as capital N goes to infinity. 01:06:19.020 --> 01:06:30.750 align:middle line:84% In particular, we get, as an immediate corollary, 01:06:30.750 --> 01:06:38.980 align:middle line:84% if all of the Fourier coefficients are 0, 01:06:38.980 --> 01:06:40.870 align:middle line:90% then f is 0, right? 01:06:40.870 --> 01:06:45.880 align:middle line:84% Because if I've proven this and all the Fourier coefficients 01:06:45.880 --> 01:06:49.180 align:middle line:84% are 0, then the Cesaro means are all 0. 01:06:49.180 --> 01:06:55.150 align:middle line:84% And therefore, since this is 0 converging to f, f must be 0. 01:06:55.150 --> 01:06:55.650 align:middle line:90% OK? 01:06:55.650 --> 01:06:59.070 align:middle line:84% And therefore, the set of exponentials-- 01:06:59.070 --> 01:07:05.770 align:middle line:84% normalized, of course-- form a maximal orthonormal subset 01:07:05.770 --> 01:07:08.650 align:middle line:90% of L2, i.e. 01:07:08.650 --> 01:07:13.210 align:middle line:84% that they're an orthonormal basis for big L2, which 01:07:13.210 --> 01:07:15.910 align:middle line:84% answers the question we had about Fourier series converging 01:07:15.910 --> 01:07:23.030 align:middle line:90% to a function in L2, OK? 01:07:23.030 --> 01:07:23.530 align:middle line:90% All right. 01:07:23.530 --> 01:07:33.070 align:middle line:84% So we'll do this just as a standard epsilon n argument. 01:07:33.070 --> 01:07:37.090 align:middle line:90% Let epsilon-- so let f be in L2. 01:07:37.090 --> 01:07:41.420 align:middle line:90% 01:07:41.420 --> 01:07:42.990 align:middle line:90% Let epsilon be positive. 01:07:42.990 --> 01:07:48.010 align:middle line:90% 01:07:48.010 --> 01:07:57.750 align:middle line:84% So we know that the continuous 2 pi periodic functions 01:07:57.750 --> 01:08:00.570 align:middle line:84% are dense in L2, because we did this in the assignment 01:08:00.570 --> 01:08:03.360 align:middle line:84% that for any f in L2 over an interval, 01:08:03.360 --> 01:08:06.480 align:middle line:84% I can find a continuous function that vanishes at the endpoints 01:08:06.480 --> 01:08:11.070 align:middle line:84% and therefore is periodic, which is close to f in L2. 01:08:11.070 --> 01:08:26.170 align:middle line:84% So there exists a g that's continuous 2 pi 01:08:26.170 --> 01:08:40.800 align:middle line:84% periodic such that f minus g in L2 norm is less than 01:08:40.800 --> 01:08:41.850 align:middle line:90% or epsilon over 3. 01:08:41.850 --> 01:08:45.540 align:middle line:90% 01:08:45.540 --> 01:09:06.210 align:middle line:84% So since sigma N g converges to g uniformly on minus pi to pi, 01:09:06.210 --> 01:09:09.840 align:middle line:84% there exists a natural number M such 01:09:09.840 --> 01:09:15.359 align:middle line:84% that for all N bigger than or equal to M, 01:09:15.359 --> 01:09:22.160 align:middle line:84% for all x minus pi to pi, I have that sigma N 01:09:22.160 --> 01:09:28.990 align:middle line:84% g of x minus g of x is less than epsilon 01:09:28.990 --> 01:09:31.060 align:middle line:90% over 3 square root of 2 pi. 01:09:31.060 --> 01:09:34.899 align:middle line:90% 01:09:34.899 --> 01:09:35.399 align:middle line:90% OK? 01:09:35.399 --> 01:09:38.939 align:middle line:90% 01:09:38.939 --> 01:09:42.917 align:middle line:84% Now, we go about the part where we replace f with g, OK? 01:09:42.917 --> 01:09:48.529 align:middle line:90% 01:09:48.529 --> 01:09:53.720 align:middle line:84% Then, for all N bigger than or equal to M, 01:09:53.720 --> 01:10:02.950 align:middle line:84% if I look at the L2 norm of sigma N of f minus f in L2, 01:10:02.950 --> 01:10:04.690 align:middle line:90% and I apply the triangle-- 01:10:04.690 --> 01:10:07.900 align:middle line:84% I add and subtract terms and apply the triangle inequality-- 01:10:07.900 --> 01:10:15.610 align:middle line:84% I get that this is less than or equal to sigma N of f minus g 2 01:10:15.610 --> 01:10:31.260 align:middle line:84% plus sigma N of g minus g in L2 plus g minus f in L2, OK? 01:10:31.260 --> 01:10:34.380 align:middle line:84% So sigma N of f minus g is equal to sigma N 01:10:34.380 --> 01:10:36.120 align:middle line:90% of f minus sigma N of g. 01:10:36.120 --> 01:10:41.110 align:middle line:84% So I use that there without explicitly stating that. 01:10:41.110 --> 01:10:48.960 align:middle line:84% So let me say sigma N of f minus sigma N of g. 01:10:48.960 --> 01:10:50.340 align:middle line:84% Just from the definition, you can 01:10:50.340 --> 01:10:58.720 align:middle line:84% check this is equal to sigma N of f minus g, OK? 01:10:58.720 --> 01:11:02.860 align:middle line:84% Now, by the bound I just proved, the L2 norm of the Cesaro mean 01:11:02.860 --> 01:11:07.190 align:middle line:84% is less than or equal to the L2 norm of the function here. 01:11:07.190 --> 01:11:10.570 align:middle line:84% So this is less than or equal to f minus g2. 01:11:10.570 --> 01:11:14.890 align:middle line:84% And then, I also have this L2 norm of f minus G there. 01:11:14.890 --> 01:11:17.170 align:middle line:90% So I'll put a 2 there plus-- 01:11:17.170 --> 01:11:20.590 align:middle line:84% and I'll actually write out what this is-- 01:11:20.590 --> 01:11:32.090 align:middle line:84% sigma N g of x minus g of x squared dx 1/2, OK? 01:11:32.090 --> 01:11:39.120 align:middle line:84% Now, f minus g is less than epsilon over 3 in L2 norm. 01:11:39.120 --> 01:11:43.190 align:middle line:84% So this is less than twice epsilon over 3. 01:11:43.190 --> 01:11:48.470 align:middle line:84% Sigma N g minus g is less than epsilon over 3 square root 01:11:48.470 --> 01:11:51.740 align:middle line:90% of 2 pi here. 01:11:51.740 --> 01:11:55.100 align:middle line:90% So I get epsilon over 3-- 01:11:55.100 --> 01:12:02.930 align:middle line:84% that pulls all the way out-- minus pi pi 1 over 2 pi dx. 01:12:02.930 --> 01:12:09.080 align:middle line:84% And I just get epsilon in the end, OK? 01:12:09.080 --> 01:12:17.480 align:middle line:90% 01:12:17.480 --> 01:12:19.490 align:middle line:90% OK. 01:12:19.490 --> 01:12:23.420 align:middle line:84% So that concludes what I wanted to do for Fourier series, 01:12:23.420 --> 01:12:29.160 align:middle line:84% at least for now, which applies what we've 01:12:29.160 --> 01:12:35.755 align:middle line:84% done for Lebesgue integration, these big LP spaces, and also 01:12:35.755 --> 01:12:38.130 align:middle line:84% some of this general machinery we've built up for Hilbert 01:12:38.130 --> 01:12:41.640 align:middle line:84% spaces to actually answer a more concrete question 01:12:41.640 --> 01:12:46.660 align:middle line:84% rather than just trying to prove general statements. 01:12:46.660 --> 01:12:48.745 align:middle line:84% General statements are very, very useful. 01:12:48.745 --> 01:12:49.870 align:middle line:90% I'm not saying they're not. 01:12:49.870 --> 01:12:52.900 align:middle line:84% But I'm just saying so that you can see a concrete problem why 01:12:52.900 --> 01:12:55.528 align:middle line:84% one would want and use functional analysis 01:12:55.528 --> 01:12:56.320 align:middle line:90% in the first place. 01:12:56.320 --> 01:12:58.940 align:middle line:90% 01:12:58.940 --> 01:13:11.380 align:middle line:84% Now, coming back to what we've done so far, 01:13:11.380 --> 01:13:20.660 align:middle line:84% so let me just make a couple of remarks 01:13:20.660 --> 01:13:21.995 align:middle line:90% about what we haven't shown. 01:13:21.995 --> 01:13:24.830 align:middle line:90% 01:13:24.830 --> 01:13:29.390 align:middle line:84% It's a very deep theorem due to Carleson. 01:13:29.390 --> 01:13:32.370 align:middle line:84% So what we've shown is that the partial sums-- 01:13:32.370 --> 01:13:37.080 align:middle line:84% so we showed the set of exponentials normalized, 01:13:37.080 --> 01:13:39.120 align:middle line:90% or a maximal orthonormal set-- 01:13:39.120 --> 01:13:41.080 align:middle line:84% I mean that they're orthonormal basis. 01:13:41.080 --> 01:13:48.520 align:middle line:84% So the partial sums converge to f in L2. 01:13:48.520 --> 01:13:52.200 align:middle line:90% So this is what we've shown. 01:13:52.200 --> 01:13:56.900 align:middle line:90% 01:13:56.900 --> 01:14:07.510 align:middle line:84% For all f in L2, the partial sums converge to f in L2, 01:14:07.510 --> 01:14:09.590 align:middle line:90% all right? 01:14:09.590 --> 01:14:13.400 align:middle line:84% But this does not translate into a point-wise statement. 01:14:13.400 --> 01:14:16.370 align:middle line:84% This does not say that the partial sums converge 01:14:16.370 --> 01:14:18.720 align:middle line:90% to f almost everywhere. 01:14:18.720 --> 01:14:19.220 align:middle line:90% OK? 01:14:19.220 --> 01:14:20.780 align:middle line:84% There is a general theorem one can 01:14:20.780 --> 01:14:26.420 align:middle line:84% say that is covered in more advanced measure theory classes 01:14:26.420 --> 01:14:29.960 align:middle line:84% where one can say that there exists a subsequence converging 01:14:29.960 --> 01:14:31.010 align:middle line:90% to f almost everywhere. 01:14:31.010 --> 01:14:36.290 align:middle line:84% But that's not very good, or at least very clean. 01:14:36.290 --> 01:14:41.970 align:middle line:84% Now, for a long time, it was not necessarily believed 01:14:41.970 --> 01:14:46.320 align:middle line:84% that the partial sums converged to f almost everywhere. 01:14:46.320 --> 01:14:59.160 align:middle line:84% But a theorem due to Carleson shows that for all f in L2, 01:14:59.160 --> 01:15:08.310 align:middle line:84% partial sums do converge to the function almost everywhere, OK? 01:15:08.310 --> 01:15:10.990 align:middle line:90% This is, in fact-- 01:15:10.990 --> 01:15:12.450 align:middle line:90% maybe this is true. 01:15:12.450 --> 01:15:13.200 align:middle line:90% Maybe this is not. 01:15:13.200 --> 01:15:14.550 align:middle line:90% I heard this from my advisor. 01:15:14.550 --> 01:15:17.430 align:middle line:84% Carleson spent a few decades trying 01:15:17.430 --> 01:15:22.490 align:middle line:84% to prove the negation of the statement, 01:15:22.490 --> 01:15:25.310 align:middle line:84% trying to come up with an example of a function whose 01:15:25.310 --> 01:15:28.070 align:middle line:84% partial sums converge don't converge almost everywhere 01:15:28.070 --> 01:15:29.480 align:middle line:90% back to the function. 01:15:29.480 --> 01:15:33.083 align:middle line:84% And then, he came up with the bright idea 01:15:33.083 --> 01:15:34.500 align:middle line:84% that, well, maybe that's not true. 01:15:34.500 --> 01:15:41.390 align:middle line:84% Let me spend some time trying to put myself in the other shoes. 01:15:41.390 --> 01:15:43.850 align:middle line:84% And within a year or a couple of years, 01:15:43.850 --> 01:15:48.460 align:middle line:84% he was able to prove this theorem, OK? 01:15:48.460 --> 01:15:54.710 align:middle line:84% So this is Carleson's theorem that we do have 01:15:54.710 --> 01:15:56.030 align:middle line:90% convergence almost everywhere. 01:15:56.030 --> 01:15:58.490 align:middle line:84% Now, you can also ask, what about convergence? 01:15:58.490 --> 01:16:01.100 align:middle line:84% So this convergence in L2 of the partial sums. 01:16:01.100 --> 01:16:05.030 align:middle line:90% We have other LP spaces, right? 01:16:05.030 --> 01:16:06.800 align:middle line:90% What about in those LP spaces? 01:16:06.800 --> 01:16:08.930 align:middle line:90% Can I replace this 2 with p? 01:16:08.930 --> 01:16:13.880 align:middle line:84% The Fourier coefficients and partial sums-- these 01:16:13.880 --> 01:16:19.830 align:middle line:84% all make sense for any big LP space. 01:16:19.830 --> 01:16:24.680 align:middle line:90% So what is known is that also-- 01:16:24.680 --> 01:16:30.350 align:middle line:84% and now, the name is escaping me, but I'll just state it. 01:16:30.350 --> 01:16:35.290 align:middle line:84% For all p between 1 and infinity, 01:16:35.290 --> 01:16:41.605 align:middle line:84% the partial sums converge to the function in lp. 01:16:41.605 --> 01:16:44.510 align:middle line:90% 01:16:44.510 --> 01:16:50.180 align:middle line:84% When p equals 1, this is false, OK? 01:16:50.180 --> 01:16:53.090 align:middle line:84% And when p equals infinity, this is also false, 01:16:53.090 --> 01:16:57.200 align:middle line:84% because the partial sums-- these are a finite linear combination 01:16:57.200 --> 01:17:01.470 align:middle line:84% of exponentials, and therefore continuous function, OK? 01:17:01.470 --> 01:17:04.430 align:middle line:84% So you can't have, for an arbitrary function in L 01:17:04.430 --> 01:17:06.380 align:middle line:84% infinity-- which can be discontinuous, 01:17:06.380 --> 01:17:08.510 align:middle line:90% just has to be bounded-- 01:17:08.510 --> 01:17:12.560 align:middle line:84% these converging to L in such a function. 01:17:12.560 --> 01:17:18.440 align:middle line:84% Because then, the limit would have to be continuous, OK? 01:17:18.440 --> 01:17:20.580 align:middle line:84% The uniform limit of continuous functions, 01:17:20.580 --> 01:17:26.860 align:middle line:84% which L infinity kind of is, has to be continuous, OK? 01:17:26.860 --> 01:17:29.590 align:middle line:84% So that's why you wouldn't expect it for L infinity. 01:17:29.590 --> 01:17:34.240 align:middle line:84% And for what one would call duality, 01:17:34.240 --> 01:17:37.720 align:middle line:84% because infinity is the dual of L1, 01:17:37.720 --> 01:17:40.090 align:middle line:90% you also don't get p equals 1. 01:17:40.090 --> 01:17:41.770 align:middle line:84% But in fact, things are worse there. 01:17:41.770 --> 01:17:43.820 align:middle line:84% You can come up with an L1 function. 01:17:43.820 --> 01:17:47.290 align:middle line:90% So that the Fourier series-- 01:17:47.290 --> 01:17:49.300 align:middle line:84% I don't think I'm lying when I say this, but-- 01:17:49.300 --> 01:18:03.580 align:middle line:84% diverges almost everywhere, I want to say, OK? 01:18:03.580 --> 01:18:05.410 align:middle line:90% I don't think I'm lying. 01:18:05.410 --> 01:18:09.010 align:middle line:84% But if p equals 1, one can come up with an example 01:18:09.010 --> 01:18:14.720 align:middle line:84% where the partial sums diverge point-wise almost everywhere, 01:18:14.720 --> 01:18:15.220 align:middle line:90% OK? 01:18:15.220 --> 01:18:19.350 align:middle line:90% 01:18:19.350 --> 01:18:19.920 align:middle line:90% OK. 01:18:19.920 --> 01:18:27.210 align:middle line:84% But to prove this flavor of statements 01:18:27.210 --> 01:18:33.870 align:middle line:84% requires deeper harmonic analysis, 01:18:33.870 --> 01:18:37.970 align:middle line:84% harmonic analysis being the umbrella 01:18:37.970 --> 01:18:46.210 align:middle line:84% that Fourier analysis sits in, and requires a knowledge of, 01:18:46.210 --> 01:18:50.140 align:middle line:84% or at least working with certain operators, which 01:18:50.140 --> 01:18:53.500 align:middle line:84% are called singular integral operators, which were developed 01:18:53.500 --> 01:18:57.570 align:middle line:84% back in the last century, middle of the last century 01:18:57.570 --> 01:19:04.130 align:middle line:84% at the University of Chicago by my mathematical grandfather 01:19:04.130 --> 01:19:08.640 align:middle line:84% and great grandfather, which gives you 01:19:08.640 --> 01:19:12.780 align:middle line:84% some beautiful results about, again, convergence of Fourier 01:19:12.780 --> 01:19:15.780 align:middle line:84% series, but also some applications to PDEs, which 01:19:15.780 --> 01:19:20.590 align:middle line:84% were why they were originally created in the first place 01:19:20.590 --> 01:19:22.270 align:middle line:90% and so on. 01:19:22.270 --> 01:19:24.090 align:middle line:84% But perhaps you'll encounter that if you 01:19:24.090 --> 01:19:28.800 align:middle line:84% take a class in harmonic analysis or Fourier series. 01:19:28.800 --> 01:19:30.550 align:middle line:84% I haven't taught the Fourier series class, 01:19:30.550 --> 01:19:32.230 align:middle line:90% so I don't know what it's about. 01:19:32.230 --> 01:19:39.390 align:middle line:84% But that kind of material will not be covered in this class. 01:19:39.390 --> 01:19:44.400 align:middle line:84% And this will be as far as we go as far as these types 01:19:44.400 --> 01:19:46.290 align:middle line:90% of questions, all right? 01:19:46.290 --> 01:20:01.180 align:middle line:84% So next time, we'll move on to minimizers over closed convex 01:20:01.180 --> 01:20:07.530 align:middle line:84% sets and consequences of that, one 01:20:07.530 --> 01:20:10.230 align:middle line:84% being that we can identify-- which 01:20:10.230 --> 01:20:13.260 align:middle line:84% is the most important application-- we can identify 01:20:13.260 --> 01:20:17.670 align:middle line:84% the dual of a Hilbert space with the Hilbert space 01:20:17.670 --> 01:20:19.950 align:middle line:90% in a canonical way. 01:20:19.950 --> 01:20:23.310 align:middle line:84% You can already prove that if you 01:20:23.310 --> 01:20:29.610 align:middle line:84% wish using the fact that every Hilbert space is asymmetrically 01:20:29.610 --> 01:20:31.230 align:middle line:90% isomorphic to little l2. 01:20:31.230 --> 01:20:35.870 align:middle line:84% You know that the dual of little l2 is 1 over q, 01:20:35.870 --> 01:20:39.150 align:middle line:84% is lq, where 1 over 2 plus 1 over q equals 1. 01:20:39.150 --> 01:20:40.500 align:middle line:90% And therefore, q equals 2. 01:20:40.500 --> 01:20:42.720 align:middle line:84% So little l2 is a dual of itself. 01:20:42.720 --> 01:20:45.280 align:middle line:90% 01:20:45.280 --> 01:20:53.010 align:middle line:84% But we'll prove it for general Hilbert spaces, which 01:20:53.010 --> 01:20:59.530 align:middle line:84% has some very important and interesting consequences when 01:20:59.530 --> 01:21:04.760 align:middle line:84% it comes to now studying, solving equations 01:21:04.760 --> 01:21:08.060 align:middle line:84% in Hilbert spaces, meaning you have linear operators. 01:21:08.060 --> 01:21:11.060 align:middle line:84% When can you solve equations involving 01:21:11.060 --> 01:21:13.160 align:middle line:84% these linear operators, and so on? 01:21:13.160 --> 01:21:16.390 align:middle line:90% All right, so we'll stop there. 01:21:16.390 --> 01:21:23.000 align:middle line:90%