Maximal Operators, Littlewood-Paley Theory, and Wavelet Approximation
Math 758L, Fall 1999
MonWed 3:45 - 5:00 p.m. , LeConte 401A
Course Topics
This course is the first semester of a year long sequence
to study maximal functions, singular integral operators, and
Fourier multipliers for classical function spaces (Lebesgue
Lp, Lorentz Lp,q, and Besov spaces).
Littlewood-Paley theory enters here in a decisive way and leads to
multiresolution representations for the classical function spaces and to
efficient wavelet approximations of their members.
See the link (Math 758S)
for a description of the second half of the course. Required topics of Fourier
analysis, interpolation theory of operators, and functional analysis
are presented as needed.
Prerequisties:
Real Analysis (Math 703-704)
Lectures:
Link to Weekly Outline
Primary References:
- E.M. Stein, "Singular integrals and differentiability properties of functions,"
Princeton University Press, Princeton, 1970.
- C. Bennett and R. Sharpley, "Interpolation of Operators," Academic Press,
New York, 1988.
Additional References:
- R.A. DeVore, Nonlinear Analysis, Acta Numerica, 7(1998),
51-150.
- E.M. Stein and G.L. Weiss, "Introduction to Fourier Analysis in Euclidean
Spaces", Princeton University Press, Princeton, 1971. [QA403. S79]
- E.M. Stein, "Harmonic Analysis: Real variable methods, orthogonality,
and oscillatory integrals," Princeton University Press, Princeton, 1993.
- E. Hernandez and G.L. Weiss, "A First Course in Wavelets," CRC Press,
New York, 1996. [QA403.3 .H47 1996]
- Kashin, B. S. and A.A. Saakyan, "Orthogonal Series," American Mathematical
Society, Providence, RI, 1989. [QA404.5 .K3413 1989]
- 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]
Course Grading
A term paper based on independent readings on topics
agreed upon by the student and instructor.
For further information, please contact
sharpley@math.sc.edu.
Last modified: 07 November 1999