Richard E. Phillips Lecture Series

The Non-linear Fourier Transform

Monday April 5, 2004 at 4:10 p.m. in B106 Wells Hall

Abstract

The non-linear Fourier transform, also known as the scattering transform, transforms potentials on the real line (or the integers) to matrix-valued functions on the real line (or on the circle). It is related to (certain) integrable systems in much the same way the linear Fourier transform is related to constant-coefficient linear PDE, and also shows up in the study of orthogonal polynomials and Jacobi matrices, among other places. Here we view the transform as a non-linear analogue of the ordinary Fourier transform, and discuss the following general question: what results of linear Fourier analysis continue to hold in the non-linear setting? We survey some progress on this question and some outstanding open problems.

Pointwise Convergence For The Linera and Non-linear Fourier Transforms

Tuesday, April 6, 2004 at 4:10 p.m. in B108 Wells Hall

Abstract

A famous theorem of Carleson asserts that the partial Fourier integrals of a square-integrable function converge for almost every choice of frequency. The no-liner analogue of this would be that the eigenfounctions of a one-dimensional Schrodinger operator with a square integrable function are bounded for almost every choice of energy. This conjecture, which would imply Carleson’s theorem in the linearized limit, remains open, although there has been some recent progress. We describe some recent progress on this problem, including a dyadic “Walsh” model of both the linear and anon-linear problems for which the analogue of Carleson’s theorem has been proven in both settings.

Invertibility of the Non-linear Fourier Transform, and A Riemann-Hilbert Problem

Wednesday, April 7, 2004 at 4:10 p.m. in B108 Wells Hall

Abstract

The non-linear Fourier transform (on the integers) enjoys several of the properties that the linear Fourier transform does, for instance there is an analogue of Plancherel’s theorem, and of the Paley-Wiener theorems. However, there is an interesting failure of invertibility (first observed by Volberg and Yuditskii) of the non-linear Fourier transform once one works with functions that are merely sqaure0integrable, which is associated with the non-uniqueness of a certain Riemann-Hilbert problem. In the special case when the non-linear Fourier transform is a rational function, we can quantify the failure of invertibility precisely in understood, although it is also closely related to the spectral and scattering theory of a certain discrete Dirac operator.

For additional information:

Department of Mathematics
Michigan State University
East Lansing, MI 48824-1027
(517)355-9680
ginther@math.msu.edu
www.math.msu.edu/Lecture_Series