Weyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces

Weyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces

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##### Date

1989

##### Authors

Walnut, David Francis

##### Advisor

Benedetto, John J.

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##### Abstract

The theory of wavelets can be used to obtain expansions of
vectors in certain spaces. These expansions are like Fourier
series in that each vector can be written in terms of a fixed
collection of vectors in the Banach space and the coefficients
satisfy a "Plancherel Theorem" with respect to some sequence
space. In Weyl-Heisenberg expansions, the expansion vectors
(wavelets) are translates and modulates of a single vector (the
analyzing vector) .
The thesis addresses the problem of the existence and
stability of Weyl-Heisenberg expansions in the space of functions
square-integrable with respect to the measure w(x) dx for a
certain class of weights w. While the question of the existence
of such expansions is contained in more general theories, the
techniques used here enable one to obtain more general and
explicit results. In Chapter 1, the class of weights of interest is defined and
properties of these weights proven.
In Chapter 2, it is shown that Weyl-Heisenberg expansions
exist if the analyzing vector is locally bounded and satisfies a
certain global decay condition.
In Chapter 3, it is shown that these expansions persist if
the translations and modulations are not taken at regular
intervals but are perturbed by a small amount. Also, the
expansions are stable if the analyzing vector is perturbed. It is
also shown here that under more general assumptions, expansions
exist if translations and modulations are taken at any
sufficiently dense lattice of points.
Like orthonormal bases, the coefficients in Weyl-Heisenberg
expansions can be formed by the inner product of the vector being
expanded with a collection of wavelets generated by a transformed
version of the analyzing vector. In Chapter 4, it is shown that
this transformation preserves certain decay and smoothness
conditions and a formula for this transformation is given.
In Chapter 5, results on Weyl-Heisenberg expansions in the
space of square-integrable functions are presented.