Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets

✍ Scribed by Claude Gasquet, Patrick Witomski (auth.)

• Publisher: Springer-Verlag New York
• Year: 1999
• Tongue: English
• Leaves: 434
• Series: Texts in Applied Mathematics 30
• Edition: 1
• Category: Library

✦ Synopsis

From the reviews:

BULLETIN OF MATHEMATICS BOOKS

"This book should have wide appeal, from those who are just getting into the area and wish to learn mathematical foundations and applications to those who are already experienced and wish to have a reference that provides a mathematically rigorous coverage of the state of the art…The coverage is thorough but not overwhelming, perhaps because the chapters are divided into lessons, allowing the reader a chance to pause and think. The authors work clearly to impart an understanding of the theory and applications, and not just offer an encyclopedic tome."

✦ Table of Contents

Front Matter....Pages i-xviii
Front Matter....Pages 1-1
Signals and Systems....Pages 3-10
Filters and Transfer Functions....Pages 11-19
Front Matter....Pages 21-21
Trigonometric Signals....Pages 23-26
Periodic Signals and Fourier Series....Pages 27-37
Pointwise Representation....Pages 39-50
Expanding a Function in an Orthogonal Basis....Pages 51-56
Frequencies, Spectra, and Scales....Pages 57-62
Front Matter....Pages 63-63
The Discrete Fourier Transform....Pages 65-73
A Famous, Lightning-Fast Algorithm....Pages 75-84
Using the FFT for Numerical Computations....Pages 85-94
Front Matter....Pages 95-95
From Riemann to Lebesgue....Pages 97-100
Measuring Sets....Pages 101-109
Integrating Measurable Functions....Pages 111-120
Integral Calculus....Pages 121-130
Front Matter....Pages 131-131
Function Spaces....Pages 133-140
Hilbert Spaces....Pages 141-152
Front Matter....Pages 153-153
The Fourier Transform of Integrable Functions....Pages 155-161
The Inverse Fourier Transform....Pages 163-170
The Space ℒ (ℝ)....Pages 171-175
The Convolution of Functions....Pages 177-185
Front Matter....Pages 153-153
Convolution, Derivation, and Regularization....Pages 187-192
The Fourier Transform on L 2 (ℝ)....Pages 193-199
Convolution and the Fourier Transform....Pages 201-207
Front Matter....Pages 209-209
Applications to Analog Filters Governed by a Differential Equation....Pages 211-219
Examples of Analog Filters....Pages 221-232
Front Matter....Pages 233-233
Where Functions Prove to Be Inadequate....Pages 235-242
What Is a Distribution?....Pages 243-250
Elementary Operations on Distributions....Pages 251-264
Convergence of a Sequence of Distributions....Pages 265-274
Primitives of a Distribution....Pages 275-279
Front Matter....Pages 281-281
The Fourier Transform of Distributions....Pages 283-296
Convolution of Distributions....Pages 297-309
Convolution and the Fourier Transform of Distributions....Pages 311-316
Front Matter....Pages 317-317
Filters, Differential Equations, and Distributions....Pages 319-324
Realizable Filters and Differential Equations....Pages 325-332
Front Matter....Pages 333-333
Periodic Distributions....Pages 335-342
Sampling Signals and Poisson’s Formula....Pages 343-352
The Sampling Theorem and Shannon’s Formula....Pages 353-363
Discrete Filters and Convolution....Pages 365-374
The z -Transform and Discrete Filters....Pages 375-381
Front Matter....Pages 383-383
The Windowed Fourier Transform....Pages 385-394
Wavelet Analysis....Pages 395-431
Back Matter....Pages 433-442

✦ Subjects

Analysis; Computational Intelligence; Math. Applications in Chemistry