MATH 758 S
Wavelet and Multiresolution Analysis Office: LeConte 313 D Office Hours: MWF 1011 a.m. and by appointment. 


Course Topics
The course will develop the basic principles and methods of (1) Fourier transforms on
R^{d}, (2) wavelets, and (3) multiresolution analysis.
Application of the concepts will be to partial differential
equations, data compression, signal and image processing.
Lectures will drawn from several general references (listed below) and
will include the following topics:
An introduction to multiresolution analysis and a full development for the Haar and Stromberg wavelets in L^{p}(R) (1 < p < ∞); Fourier analysis on R^{d}, the L^{1} theory and computation of transforms for the Gaussian, Poisson, and heat kernels; the L^{2} theory and Plancherel theorem; the L^{p} theory and Hilbert transform (both as multiplier and singular integral); Heisenberg inequality, Poisson summation formula; bandlimited functions and the PaleyWiener Theorem; the sinc function and ShannonWhittacker Sampling Theorem; development of the general MeyerMallat multiresolution analysis, scaling function and construction of wavelets; wavelets and Fourier transform criteria, Riesz's factorization lemma, Daubechies' construction of smooth compactly supported wavelets; Rademacher functions, probabilistic independence and unconditional convergence of series in Banach spaces, Khinchin's inequality, rearrangement and multipliers of series, wavelet bases as unconditional bases; DeVore's nonlinear approximation and Besov spaces; optimal entropy encoding in L^{p}; redundant approximation using Courant elements and surface compression.
Prerequisites
Fourier Analysis
(Math 750)
Lectures:
Link to Weekly Outline
General References
Course Grading
A term paper and hour lecture on a topic negotiated with the instructor.


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