Lecture 27 MIT 3.016 (Fall 2005) © W. Craig Carter 2003-2005

Basis and Orthogonal Functions

This is a generalization of the Fourier series. The orthogonality of the trignometric functions
(i.e., $\int_{0}^{1}$ Sin[2 π n x] Sin[2 π m x]dx =

$\frac{1}{2}$	if n=m
0	if n≠m

)
was the key to the ``trick''  that permitted calculation of the Fourier coefficients.
This is a generalization of dot or scalar vector product, but for functions.  The norm that is defined above for the trignometric functions is fairly simple, it is the l2-norm:
f(x) • g(x) ≡  ∫ f(x) g(x) dx        (l2-norm)
However, there can be many other types of norms, for instance a Gaussian weighted l2-norm
f(x) • g(x) ≡  ∫ f(x) g(x)

ⅇ^{- x^2}

dx (Gaussian weighted l2-norm)
and many many others.

Both the norm and the domain of integration must be defined.

To represent a function in terms of a sum of basis functions with coefficients, a orthogonality relation must be obtained for the basis functions.
We will do a few examples of orthogonality relations, the coefficients can be found as a straightforward extension of the Fourier method.

Establish by example the orthogonality relations for the Lengendre Polynomials

In[115]:=

$Table [Integrate [LegendreP [i, x] LegendreP [5, x], {x, - 1, 1}], {i, - 10, 10}]$

Out[115]=

${0, 0, 0, 0, \frac{2}{11}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \frac{2}{11}, 0, 0, 0, 0, 0}$

Calculate the norm as a function of the order of the Legendre polynomial

In[116]:=

$Table [Integrate [LegendreP [i, x] LegendreP [i, x], {x, - 1, 1}], {i, 0, 15}]$

Out[116]=

${2, \frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}, \frac{2}{13}, \frac{2}{15}, \frac{2}{17}, \frac{2}{19}, \frac{2}{21}, \frac{2}{23}, \frac{2}{25}, \frac{2}{27}, \frac{2}{29}, \frac{2}{31}}$

Make a guess at what the norm looks like

In[117]:=

$Table [\frac{2}{2 i + 1}, {i, 0, 15}]$

Out[117]=

The orthogonality relation is plausibly: $\int_{- 1}^{1}$ LegendreP[i,x] LegendreP[j,x]dx =

$\frac{2}{2 i + 1}$	if \| i\|=\|j\|
0	if \|i \|≠\|j\|

In fact, this is correct

Representation of a function in terms of the Legendre basis

This demonstrates how to take a specific set (basis) of orthogonal functions and represent an arbitrary function in terms of an infinite sum of terms involving the basis functions, each with its own coefficient. This is a direct analog to what we did with Fourier series.

A function to calculate the coefficients

In[118]:=

$LegendreCoefficient [f_, i_] := \frac{2 i + 1}{2} Integrate [f [x] LegendreP [i, x], {x, - 1, 1}]$

Use Legendre functions to represent the function $ⅇ^{- x^{2}}$

This is an example function:

In[119]:=

$AFunc [x_] := Exp [- x^2]$

Calculate the first twenty coefficients

In[120]:=

$Lcofs = Table [LegendreCoefficient [AFunc, i], {i, 0, 20}]$

Out[120]=

${\frac{1}{2} \sqrt{π} Erf [1], 0, \frac{5 (- 6 + ⅇ \sqrt{π} Erf [1])}{8 ⅇ}, 0, \frac{9 (- 230 + 57 ⅇ \sqrt{π} Erf [1])}{64 ⅇ}, 0, \frac{13 (- 7938 + 1955 ⅇ \sqrt{π} Erf [1])}{256 ⅇ}, 0, \frac{17 (- 1579218 + 388955 ⅇ \sqrt{π} Erf [1])}{4096 ⅇ}, 0, \frac{147 (- 14740550 + 3630537 ⅇ \sqrt{π} Erf [1])}{16384 ⅇ}, 0, \frac{25 (- 16734510494 + 4121641293 ⅇ \sqrt{π} Erf [1])}{131072 ⅇ}, 0, \frac{29 (- 1621927519250 + 399474095163 ⅇ \sqrt{π} Erf [1])}{524288 ⅇ}, 0, \frac{33 (- 1463595204914978 + 360477495590355 ⅇ \sqrt{π} Erf [1])}{16777216 ⅇ}, 0, \frac{37 (- 188369607025634322 + 46394661555265595 ⅇ \sqrt{π} Erf [1])}{67108864 ⅇ}, 0, \frac{41 (- 54487481501236728090 + 13420043196802227271 ⅇ \sqrt{π} Erf [1])}{536870912 ⅇ}}$

Construct a vector of the eigenfunctions

In[121]:=

$Lfuncs = Table [LegendreP [i, x], {i, 0, 20}];$

$General :: spell1 : Possible spelling error: new symbol name \" Lfuncs \" is similar to existing symbol \" funcs \". More…$

Visualize the approximation of the first twenty terms

In[122]:=

$Plot [{Lcofs . Lfuncs, Exp [- x^2]}, {x, - 1, 1}, PlotStyle \to {{Hue [0], Thickness [0.01]}, {Hue [0.66], Thickness [0.005]}}]$

[Graphics:HTMLFiles/Lecture-27_1.gif]

Out[122]=

$⁃ Graphics ⁃$

Try another function

In[123]:=

$AFunc [x_] := Sin [5 x] Sin [(5 + \frac{1}{10}) x]$

Coefficients out to 10 terms

In[124]:=

$Lcofs = Table [LegendreCoefficient [AFunc, i], {i, 0, 10}]$

Out[124]=

${\frac{1}{2} (10 Sin [\frac{1}{10}] - \frac{10}{101} Sin [\frac{101}{10}]), 0, \frac{5}{2} (300 Cos [\frac{1}{10}] - \frac{300 Cos [\frac{101}{10}]}{10201} - 2990 Sin [\frac{1}{10}] - \frac{99010 Sin [\frac{101}{10}]}{1030301}), 0, \frac{9}{2} (- 1049000 Cos [\frac{1}{10}] - \frac{9151000 Cos [\frac{101}{10}]}{104060401} + 10455010 Sin [\frac{1}{10}] - \frac{592059010 Sin [\frac{101}{10}]}{10510100501}), 0, \frac{13}{2} (10382402100 Cos [\frac{1}{10}] - \frac{100389242100 Cos [\frac{101}{10}]}{1061520150601} - 103477709990 Sin [\frac{1}{10}] + \frac{6521460203990 Sin [\frac{101}{10}]}{107213535210701}), 0, \frac{17}{2} (- 202432299296400 Cos [\frac{1}{10}] + \frac{835591477136400 Cos [\frac{101}{10}]}{10828567056280801} + 2017570746870010 Sin [\frac{1}{10}] + \frac{114086914568321990 Sin [\frac{101}{10}]}{1093685272684360901}), 0, \frac{21}{2} (6538104407577605500 Cos [\frac{1}{10}] + \frac{5563454313317994500 Cos [\frac{101}{10}]}{110462212541120451001} - 65162961832469984990 Sin [\frac{1}{10}] - \frac{1906651913045646525010 Sin [\frac{101}{10}]}{11156683466653165551101})}$

In[125]:=

$Lfuncs = Table [LegendreP [i, x], {i, 0, 10}];$

Visualize the approximation

In[126]:=

$Plot [{Lcofs . Lfuncs, AFunc [x]}, {x, - 1, 1}, PlotStyle \to {{Hue [0], Thickness [0.01]}, {Hue [0.66], Thickness [0.005]}}]$

[Graphics:HTMLFiles/Lecture-27_2.gif]

Out[126]=

$⁃ Graphics ⁃$

Created by Mathematica (November 26, 2005)