An Overview of Wavelet Theory,
With an eye on Superoscillations,
Lectures by Professor David Walnut,
George Mason University
November 6-10, 2017,
Chapman University, ORANGE, CA
David F. Walnut received his Ph.D. degree in mathematics from the University of Maryland in 1989 under the direction of John Benedetto.
He has been on the faculty at George Mason University since 1990, where he currently holds the rank of Professor.
Professor Walnut's mathematical interests include Euclidean harmonic analysis, time-frequency analysis and sampling theory.
All talks will be held in Von Neumann Hall at 545 West Palm Ave, Orange, CA 92868 SCHEDULE: November 6th from 4pm to 5:30pm Lecture 1: Some Time-Frequency Transforms
What is a Time-Frequency distribution/representation?
The Short-Time Fourier Transform
The Ambiguity Function and Wigner's Distribution
Uncertainty Principles
D. Gabor's notion of information area
Discretizing the Short-Time Fourier Transform
Stability in time-frequency representations
The Balian-Low Theorem
November 7th from 4pm to 5:30pm Lecture 2: Gabor Theory
The Balian-Low Theorem (cont'd)
The Amalgam BLT (Heil)
The Zak transform and proofs.
Lecture 3: Theory of Frames
Finite Frames
Abstract Frames
The Frame operator
Riesz bases
Some historical remarks
The BLT from the perspective of frames
November 8th from 4pm to 7pm Lecture 4: Structure Theorems for Gabor Frames
Existence of Gabor frames
Zak transform methods
Density theorems for Gabor frames
The Wexler-Raz and Ron-Shen duality
Wilson bases
Lecture 5: Continuous and Discrete Wavelets
The Continuous Wavelet transform of Grossman and Morlet
The CWT as a time-frequency (time-scale) transformation
Relation to the Calderon Reproducing Formula
Discrete Wavelet decompositions of Frazier and Jawerth
Relation to Littlewood-Paley theory
November 9th from 4pm tp 7pm Lecture 6: Orthonormal Bases of Wavelets
Representations of the Heisenberg group and the Affine group
Co-orbit spaces from irreducible group representations
Co-orbit spaces as reproducing kernel Hilbert spaces
Frames and sampling in co-orbit spaces
November 10th from 2pm to 3pm Lecture 9: Wavelets in Functional Analysis
Modulation spaces
The Feichtinger Algebra
Pseudodifferential operators and Gabor frames
Wavelets as unconditional bases for Banach spaces
Wavelets and operators
About Professor David Walnut:
David F. Walnut received his Ph.D. degree in mathematics from the University of Maryland, USA in 1989 under the direction of John Benedetto. He has been on the faculty at George Mason University in Fairfax, Virginia, USA since 1990, where he currently holds the rank of Professor. His mathematical interests include Euclidean harmonic analysis, time-frequency analysis and sampling theory.
Lecture Slides: Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 CECHA