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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Introduction to Fourier Analysis and Wavelets (Hardcover)
Loot Price: R3,685
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Introduction to Fourier Analysis and Wavelets (Hardcover)
Series: Graduate Studies in Mathematics
Expected to ship within 12 - 17 working days
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This book provides a concrete introduction to a number of topics in
harmonic analysis, accessible at the early graduate level or, in
some cases, at an upper undergraduate level. Necessary
prerequisites to using the text are rudiments of the Lebesgue
measure and integration on the real line. It begins with a thorough
treatment of Fourier series on the circle and their applications to
approximation theory, probability, and plane geometry (the
isoperimetric theorem). Frequently, more than one proof is offered
for a given theorem to illustrate the multiplicity of approaches.
The second chapter treats the Fourier transform on Euclidean
spaces, especially the author's results in the three-dimensional
piecewise smooth case, which is distinct from the classical Gibbs -
Wilbraham phenomenon of one-dimensional Fourier analysis. The
Poisson summation formula treated in Chapter 3 provides an elegant
connection between Fourier series on the circle and Fourier
transforms on the real line, culminating in Landau's asymptotic
formulas for lattice points on a large sphere. Much of modern
harmonic analysis is concerned with the behavior of various linear
operators on the Lebesgue spaces Lp (Rn). Chapter 4 gives a gentle
introduction to these results, using the Riesz - Thorin theorem and
the Marcinkiewicz interpolation formula. One of the long-time users
of Fourier analysis is probability theory. In Chapter 5 the central
limit theorem, iterated log theorem, and Berry - Esseen theorems
are developed using the suitable Fourier-analytic tools. The final
chapter furnishes a gentle introduction to wavelet theory,
depending only on the L2 theory of the Fourier transform (the
Plancherel theorem). The basic notions of scale and location
parameters demonstrate the flexibility of the wavelet approach to
harmonic analysis. The text contains numerous examples and more
than 200 exercises, each located in close proximity to the related
theoretical material.
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