MATH 758 S
Wavelet and Multiresolution Analysis
Office: LeConte 313 D
Office Hours: MWF 10-11 a.m. and by appointment.
The course will develop the basic principles and methods of (1) Fourier transforms on Rd, (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 Lp(R) (1 < p < ∞); Fourier analysis on Rd, the L1 theory and computation of transforms for the Gaussian, Poisson, and heat kernels; the L2 theory and Plancherel theorem; the Lp theory and Hilbert transform (both as multiplier and singular integral); Heisenberg inequality, Poisson summation formula; band-limited functions and the Paley-Wiener Theorem; the sinc function and Shannon-Whittacker Sampling Theorem; development of the general Meyer-Mallat 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 Lp; redundant approximation using Courant elements and surface compression.
Fourier Analysis (Math 750)
Link to Weekly Outline
A term paper and hour lecture on a topic negotiated with the instructor.
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