# Math 133: General Course Outline

## Catalog Description

133. Introduction to Fourier Analysis. (4) Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B, 131A. Fourier series, Fourier transform in one and several variables, finite Fourier transform. Applications, in particular, to solving differential equations. Fourier inversion formula, Plancherel theorem, convergence of Fourier series, convolution. P/NP or letter grading.

## Course Information:

This syllabus is based on a single midterm; instructors who wish to give a second midterm may adjust the syllabus appropriately, or give the second midterm in section. The lecturer may also wish to expand the applications components (lectures 11-12, 22-24, 26-28) or move them earlier in the course.

Math 133 is the introduction to Fourier series, the Fourier transform in one and several variables, finite Fourier transform, applications, in particular to solving differential equations. Fourier inversion formula, Plancherel's theorem, convergence of Fourier series, convolution.

## Textbook

E. Stein and R. Shakarchi, Fourier Analysis: An Introduction (Princeton Lectures in Analysis, Volume 1), Princeton University Press.

## Schedule of Lectures

Lecture Section Topics

1

Review: Complex numbers (esp. Euler's formula); periodic functions; functions on an interval; functions on a circle; continuous functions; continuously differentiable functions; Riemann integrable functions (or at least piecewise continuous functions). No

2

Does every function have a Fourier series? Formal computation of Fourier coefficients. Inversion formula for trigonometric polynomials. Examples of Fourier series (esp. Dirichlet kernel).

3

Review of convergence, uniform convergence. Do Fourier series converge back to the original function? Injectivity of the Fourier transform for continuous functions.

4

Uniform convergence for absolutely summable Fourier coefficients. Relationship between differentiation and the Fourier transform. Uniform convergence for C^2 functions. (Optional) Some foreshadowing of future convergence results.

5

Convolutions of continuous periodic functions: examples and basic properties. Connections with Fourier coefficients. Connection between partial sums and the Dirichlet kernel.

6

Convolutions of integrable periodic functions: approximation of integrable functions by continuous ones. Approximation via convolution by good kernels.

7

Badness of the Dirichlet kernel; Gibbs' phenomenon. Cesaro means; Fejer kernel. Fejer's theorem. Uniform approximation of continuous functions by trigonometric polynomials.

8

Leeway

9

Review of vector spaces, inner product spaces, orthonormal sets, Cauchy-Schwarz inequality, Pythagoras's theorem. Orthonormality of the Fourier basis. Bessel's inequality. Best mean-square approximation by trigonometric polynomials.

10

Mean-square convergence of Fourier series for continuous functions. Mean-square convergence of Fourier series for Riemann-integrable functions. Plancherel's theorem, Parseval's theorem. Riemann-Lebesque lemma.

11-12

Applications and further properties of Fourier series, at instructor's discretion. Some suggestions: Summation of 1/n^2; local convergence of Fourier series at smooth points; smoothness of a function versus decay of Fourier coefficients; a continuous func

13

Leeway/review

14

Midterm.

15

From Fourier series to Fourier integrals - an informal discussion. Review of improper integrals. Functions of moderate decrease. Functions of rapid decrease. Schwartz functions. Definition of the Fourier transform.

16

Basic algebraic properties of the Fourier transform. Preservation of the Schwartz space.

17

Fourier transform of Gaussians. Gaussians as good kernels.

18

Multiplication formula. Fourier inversion formula. Bijectivity on Schwartz space.

19

Fourier transform and convolutions. Plancherel's theorem. Extension to functions of moderate decrease.

20-21

Integration on R^d; Fourier transform on R^d; key properties.

22-24

Applications to PDE: heat equation; Laplace's equation. (Optional) The wave equation (in 1D or higher dimensions).

25

Z_N. The finite Fourier transform; key properties.

26-28

Applications and further properties of Fourier transforms, at instructor's discretion. Some suggestions: The fast Fourier transform; fast multiplication; Heisenberg uncertainty principle; Comparison of Fourier and Laplace transforms; The Fourier-Bessel tr

29

Leeway/review.