Singular Integral Operators, Littlewood-Paley Theory, and Wavelet Approximation

Math 758S, Spring 2000
MWF   9:05 - 9:55 a.m., LeConte 316

Professor Robert Sharpley
Department of Mathematics
University of South Carolina

Course Topics
    This course is a continuation of Math 758L focusing on multiresolution representations, Jackson and Bernstein inequalities, and nonlinear wavelet approximations. In particular, characterization of classical Besov spaces in terms of moduli of continuity, approximation spaces, and their wavelet representations will be presented. Background topics of approximation theory, Fourier analysis and functional analysis will be presented as needed.

Prerequisties:
    Real Analysis (Math 703-704) and Fourier Analysis in R^d (e.g. Math 758L)

Lectures:
    Link to Weekly Outline

Basic References: (In their order of appearance)

• R.A. DeVore and V. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305(1988), 397-414.
• P. Petrushev and V. Popov, "Rational Approximation of Real Functions," Cambridge University Press, New York, 1987.
• R.A. DeVore and G.G. Lorentz, "Constructive Approximation," Grundlehren der Mathematischen Wissenschaften 303, Springer-Verlag, Berlin, 1993.
• R.A. DeVore, B. Jawerth, and V. Popov, Compression of Wavelet Coefficients, Amer. J. Math. 114(1992), 737-785.
• E. Hernandez and G.L. Weiss, "A First Course in Wavelets," CRC Press, New York, 1996. [QA403.3 .H47 1996]
• B.S. Kashin and A.A. Saakyan, "Orthogonal Series," American Mathematical Society, Providence, RI, 1989. [QA404.5 .K3413 1989]
• R.A. DeVore, Nonlinear Analysis, Acta Numerica, 7(1998), 51-150.
• A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore, Tree Approximation and Encoding, preprint.

General Background:

• C. Bennett and R. Sharpley, "Interpolation of Operators," Academic Press, New York, 1988.
• E.M. Stein, "Singular integrals and differentiability properties of functions," Princeton University Press, Princeton, 1970.

Additional Reading:

• M. Frazier, B. Jawerth, G.L. Weiss, "Littlewood-Paley Theory and the Study of Function Spaces," CBMS Regional Conference Series # 79, American Mathematical Society, 1991. [QA.R33 no. 79]
• I. Daubechies, "Ten Lectures on Wavelets," CBMS Regional Conference Series in Applied Mathematics # 61, SIAM, 1992. [QA403.3 .D38 1992]