David F. Walnut

Continuous and discrete wavelet transforms

This paper is an expository survey of results on integral representations and discrete sum expansions of functions in

Continuity properties of the Gabor frame operator

L^2 ({\bf R})

Gabor systems and the Balian-Low Theorem

in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called ’wavelets,’ which arise as translations and dilations of a single function. In each case it is shown how to represent any function in

Systems of convolution equations, deconvolution, Shannon Sampling, and the wavelet and Gabor transforms

L^2 ({\bf R})

Measurement of Time-Variant Linear Channels

as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.

Applications of Gabor and wavelet expansions to the Radon transform

Abstract For a Gabor frame with small lattice, it is shown that the frame operator is continuous and invertible on many Banach spaces defined by smoothness and decay properties. As a consequence, it is shown that there exist Gabor-type frame/dual frame pairs and tight frames with good decay and smoothness properties. Also, it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly (e.g., in Lp and Sobolev norms). Also, explicit estimates on lattice sizes for which such results hold can be obtained.

A Riesz basis for Bargmann-Fock space related to sampling and interpolation

The Balian-Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system {su2πimbt g(t — na)} m,n∈ℤ with ab = 1 forms an orthonormal basis for L 2(ℝ) then The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g′)⋀(γ) = 2πiγĝ(γ), the role of differentiation in the proof of the BLT is examined carefully. We include the construction of a complete Gabor system of the form {e 2πibmt g(t — a n )} such that {(a n ,b m )} has density strictly less than 1, and an Amalgam BLT that provides distinct restrictions on Gabor systems {e 2πimbt g(t — na)} that form exact frames.

Solutions to deconvolution equations using nonperiodic sampling

Linear translation invariant systems (e.g., sensors, linear filters) are modeled by the convolution equation

Lattice size estimates for Gabor decompositions

s = f *\mu

Sampling and Reconstruction of Operators

, where f is the input signal,