Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets

Front Cover
Springer Science & Business Media, Nov 6, 1998 - Mathematics - 442 pages
The object of this book is two-fold -- on the one hand it conveys to mathematical readers a rigorous presentation and exploration of the important applications of analysis leading to numerical calculations. On the other hand, it presents physics readers with a body of theory in which the well-known formulae find their justification. The basic study of fundamental notions, such as Lebesgue integration and theory of distribution, allow the establishment of the following areas: Fourier analysis and convolution Filters and signal analysis time-frequency analysis (gabor transforms and wavelets). The whole is rounded off with a large number of exercises as well as selected worked-out solutions.
 

Contents

Signals and Systems
3
Filters and Transfer Functions
11
Trigonometric Signals
23
Pointwise Representation
39
Expanding a Function in an Orthogonal Basis
51
Frequencies Spectra and Scales
57
The Discrete Fourier Transform
63
A Famous LightningFast Algorithm
75
Analog Filters Governed by a Differential
211
Examples of Analog Filters
221
Where Functions Prove to Be Inadequate
235
What Is a Distribution?
243
Elementary Operations on Distributions
251
Convergence of a Sequence of Distributions
265
Primitives of a Distribution
275
Convolution and the Fourier
281

Using the FFT for Numerical Computations
85
From Riemann to Lebesgue
97
Integrating Measurable Functions
111
Integral Calculus
121
Function Spaces
133
Hilbert Spaces
141
Convolution and the Fourier
153
The Inverse Fourier Transform
163
The Space SP R
171
The Convolution of Functions
177
Convolution Derivation and Regularization
187
The Fourier Transform on L2R
193
Convolution and the Fourier Transform
201
Convolution of Distributions
297
Convolution and the Fourier Transform
311
Filters Differential Equations and Distributions
319
Realizable Filters and Differential Equations
325
Periodic Distributions
335
Sampling Signals and Poissons Formula
343
The Sampling Theorem and Shannons Formula
353
Discrete Filters and Convolution
365
The zTransform and Discrete Filters
375
The Windowed Fourier Transform
385
Wavelet Analysis
395
References
433
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