Topics in Analysis

Math 684, CRN 13831

Fall, 2018

Lecture:

Instructor:

Office:

Telephone:

e-mail:

yuan@uoregon.edu

Webpage:

http://uoregon.edu/~yuan/Teaching/M684.html

Office Hours:

Course description

We will cover basics and selected topics in Fourier Analysis, Approximation Theory and Orthogonal Polynomials. We will start with the Fourier series, one of the most basis objects in analysis and the root of all topics that we will consider. Below is a list of topics that we will try to cover (in this and the next term):

Fourier series: convergence, summability, $L^p$ theory
Approximation by trigonometric polynomials
Fourer transforms and distributions
Summability of Fourier integrals
Maximal functions and Calderon-Zygmund decomposition
Multiple Fourer series and Poisson summation formula
Approximation by polynomials
Orthogonal polynomials and Fourier orthogonal series in one variable
Spherical harmonics and analysis on the unit sphere
Orthogonal polynomials and Fourier series in several variables

References

Most of the materials that we cover are classical and here is a list of reference books

Fourier Series and Harmonic Analysis

H. Dym, H. McKean: Fourier Series and Integrals. A classic with a broad view and many applications.
L. Grafakos, Classical Fourier Analysis. A more recent book, first one of two books, that covers way too many topics and details than we can contemplate.
Y. Katznelson, An Introduction to Harmonic Analysis. A classic and remain useful, especially for our first part.
T. Korner, Fourier Analysis. Easy to follow, breaking into short sections, a good book for leisure reading.
E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. A classic and a great book on the topic.
Elias M. Stein, Singular integrals and differentiability properties of functions. A classic and a great book.
A. Zygmund, Trigonometric Series. A treatise of classical harmonic analysis.

Approximation Theory and Orthogonal Polynomials

G. Andrews, R. Askey and R. Roy, Special functions}. A very useful book on special functions and orthogonal polynomials.
F. Dai and Y. Xu, Approximation theory and harmonics analysis on spheres and balls.
R. Devore and G. Lorentz, Constructive Approximation. There are numerous books on approximation theory, this one is more contemporty.
C. Dunkl and Y. Xu, Orthogonal polynomials of several variables.
G. Szego, Orthogonal polynomials. A classic, the "bible" on orthognal polynomials.