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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Integral equations
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The Hilbert Transform of Schwartz Distributions Applications (Hardcover)
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The Hilbert Transform of Schwartz Distributions Applications (Hardcover)
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This book provides a modern and up-to-date treatment of the Hilbert
transform of distributions and the space of periodic distributions.
Taking a simple and effective approach to a complex subject, this
volume is a first-rate textbook at the graduate level as well as an
extremely useful reference for mathematicians, applied scientists,
and engineers.
The author, a leading authority in the field, shares with the
reader many new results from his exhaustive research on the Hilbert
transform of Schwartz distributions. He describes in detail how to
use the Hilbert transform to solve theoretical and physical
problems in a wide range of disciplines; these include aerofoil
problems, dispersion relations, high-energy physics, potential
theory problems, and others.
Innovative at every step, J. N. Pandey provides a new definition
for the Hilbert transform of periodic functions, which is
especially useful for those working in the area of signal
processing for computational purposes. This definition could also
form the basis for a unified theory of the Hilbert transform of
periodic, as well as nonperiodic, functions.
The Hilbert transform and the approximate Hilbert transform of
periodic functions are worked out in detail for the first time in
book form and can be used to solve Laplace's equation with periodic
boundary conditions. Among the many theoretical results proved in
this book is a Paley-Wiener type theorem giving the
characterization of functions and generalized functions whose
Fourier transforms are supported in certain orthants of Rn.
Placing a strong emphasis on easy application of theory and
techniques, the book generalizes the Hilbert problem in higher
dimensions and solves it in function spaces as well as in
generalized function spaces. It simplifies the one-dimensional
transform of distributions; provides solutions to the
distributional Hilbert problems and singular integral equations;
and covers the intrinsic definition of the testing function spaces
and its topology.
The book includes exercises and review material for all major
topics, and incorporates classical and distributional problems into
the main text. Thorough and accessible, it explores new ways to use
this important integral transform, and reinforces its value in both
mathematical research and applied science.
The Hilbert transform made accessible with many new formulas and
definitions
Written by today's foremost expert on the Hilbert transform of
generalized functions, this combined text and reference covers the
Hilbert transform of distributions and the space of periodic
distributions. The author provides a consistently accessible
treatment of this advanced-level subject and teaches techniques
that can be easily applied to theoretical and physical problems
encountered by mathematicians, applied scientists, and graduate
students in mathematics and engineering.
Introducing many new inversion formulas that have been developed
and applied by the author and his research associates, the book:
* Provides solutions to the distributional Hilbert problem and
singular integral equations
* Focuses on the Hilbert transform of Schwartz distributions,
giving intrinsic definitions of the space H(D) and its topology
* Covers the Paley-Wiener theorem and provides many important
theoretical results of importance to research mathematicians
* Provides the characterization of functions and generalized
functions whose Fourier transforms are supported in certain
orthants of Rn
* Offers a new definition of the Hilbert transform of the periodic
function that can be used for computational purposes in signal
processing
* Develops the theory of the Hilbert transform of periodic
distributions and the approximate Hilbert transform of periodic
distributions
* Provides exercises at the end of each chapter--useful to
professors in planning assignments, tests, and problems
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