18.103 | Fall 2013 | Undergraduate

Fourier Analysis

Lecture Notes & Readings

The readings are assigned in the textbook
Adams, Malcolm Ritchie, and Victor Guillemin. Measure Theory and Probability. Birkhäuse, 1996. ISBN: 9780817638849. [Preview with Google Books]

Additional notes are provided for selected lectures to supplement the textbook.

1 Coin Tossing, Law of Large Numbers, Rademacher Functions Sections 1.1 and 1.2 Introductory Lecture (PDF)
2 Measure Theory, Random Models Sections 1.3 and 1.4 Boolean Rings (PDF)
3 Measurable Functions, Lebesgue Integral Sections 2.1 and 2.2 <no notes>
4 Convergence Theorems, Riemann Integrability Sections 2.3 and 2.4 <no notes>
5 Fubini’s Theorem, Independent Random Variables Sections 2.5, 2.6, and 2.7 <no notes>
6 Lebesgue Spaces, Inner Products Sections 3.1 and 3.2 Lp Theory (PDF)
7 Hilbert Space, Midterm Review Section 3.3 Hilbert Space and Orthonormal Bases (PDF)
8 Fourier Series and their Convergence Section 3.4 Fourier Series, Part 1 (PDF), Fourier Series, Part 2 (PDF)
9 Applications of Fourier Series <no readings> Fourier Series, Part 3 (PDF)
10 Fourier Integrals Section 3.5 Fourier Integrals (PDF)
11 Fourier Integrals of Measures, Central Limit Theorem Section 3.8 Fourier Integrals, Measures, and Central Limit Theorem (PDF)
12 Brownian Motion <no readings> Brownian Motion (PDF)
13 Brownian Motion Concluded, Review for Final Exam <no readings> <no notes>

Course Info

As Taught In
Fall 2013
Learning Resource Types
Problem Sets
Lecture Notes