WIENER AMALGAM SPACES IN GENERALIZED HARMONIC ANALYSIS AND WAVELET THEORY
WIENER AMALGAM SPACES IN GENERALIZED HARMONIC ANALYSIS AND WAVELET THEORY
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Date
1990
Authors
Heil, Christopher Edward
Advisor
Benedetto, John J.
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Abstract
This thesis is divided into four parts. Part I, Introduction and Notation,
describes the results contained in the thesis and their background. Part II,
Wiener Amalgam Spaces, is an expository introduction to Feichtinger's general
amalgam space theory, which is used in the remainder of the thesis to formulate
and prove results. Part III, Generalized Harmonic Analysis, presents
new results in that area. Part IV, Wavelet Theory, contains exposition and
miscellaneous results on Gabor ( also known as Weyl-Heisenberg) wavelets.
Amalgam, or mixed-norm, spaces are Banach spaces of functions determined
by a norm which distinguishes between local and global properties of
functions. Specific cases were introduced by Wiener. Feichtinger has developed
a far-reaching generalization of amalgam spaces, which allows general
function spaces norms as local or global components. We use Feichtinger's
amalgam theory, on d-dimensional Euclidean space under componentwise
multiplication, to prove that the Wiener transform (introduced by Wiener
to analyze the spectra of infinite-energy signals) is an invertible mapping of the amalgam space with local L2 and global Lq. components onto an appropriate
space defined in terms of the variation of functions, for each q between one
and infinity. As corollaries, we obtain results of Beurling on the Fourier transform
and results of Lau and Chen on the Wiener transform. Moreover, our
results are carried out in higher dimensions. In addition, we prove that the
higher-dimensional variation spaces are complete by using Masani's helices;
this generalizes a one-dimensional result of Lau and Chen.
In wavelet theory, we present a survey of frames in Hilbert and Banach
spaces and the use of the Zak transform in analyzing Gabor wavelets. Frames
are an alternative to unconditional bases in these spaces; like bases, they
Provide representations of each element of the space in terms of the frame
elements, and do so in a way in which the scalars in the representation are
explicitly known. However, unlike bases, the representations need not be
unique. We then discuss the specific case of Gabor frames in the space of
square-integrable functions, concentrating on the role of the Zak transform in
the analysis of such frames.