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

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

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]

For further information, please contact