Math 529: Functional Analysis

Introduction to Harmonic Analysis

David L Ragozin

Spring 2000, MWF 12:30-1:20

Prerequisite: Functional Analysis(Math 527/8) although its prerequisites -- Real Analysis (Math 524/5/6) or Linear Analysis (Math 554/5/6) -- will suffice for much of what we will cover.

Text:  H. Dym and H. P. McKean, Fourier Series and integrals.

Content: An introduction to both classical and modern harmonic analysis in concrete contexts with applications.  These chapters in Dym and McKean will be covered:

  1. Fourier series (development and convergence of expansions of periodic functions or functions on the circle -- R/Z or higher-dimensional tori Rn/ Zn -- in terms of exponentials exp(2ik .t) ),
  2. Fourier integrals (the non-periodic expansion and convergence questions on R or Rn),
  3. Fourier integrals and complex function theory (what happens when a function on the circle or the real line extends to a holomorphic function on some subset of the complex plane?),
  4. Fourier series and integrals on groups (topics like the preceding for functions on the rotation group O(3) of 3 x 3 orthogonal matrices and the group SL(2,R) of 2 x 2 matrices of determinant 1).
These contain many applications, such as the isoperimetric problem, Jacobi's identity for the theta function, heat flow, wave motion, random walks, the central limit theorem, Heisenberg's inequality, Minkowski's theorem in the geometry of numbers, polynomial approximation, the distribution of primes, Gauss' law of quadratic reciprocity, representations of the rotation group SO(3).

In parallel to the Dym and McKean presentation, the course will develop the perspective of Fourier series and integrals relative to operators on some class of functions on a geometric space X on which a group G acts when those operators commute with the action of the group.   Examples will include X = Sn (the n-sphere) with the action of the orthogonal matrices and any rotation invariant kernel operator or differential operator, and X = the upper half-plane with the action of SL(2,R) via linear fractional transforms and various kernel and differential operators.