Fourier Analysis and Applications: Filtering, Numerical Computation, WaveletsThe 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 |
| 433 | |
Other editions - View all
Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets Claude Gasquet,Patrick Witomski No preview available - 2013 |
Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets Claude Gasquet,Patrick Witomski No preview available - 2014 |
Common terms and phrases
additions analysis apply approximation Assume basis bounded called compact complex compute Consider constant continuous converges convolution defined Definition denote derivative differentiable discrete distribution elements equal equation example Exercise exists expression extend fact FIGURE filter finite fixed formula Fourier coefficients Fourier series Fourier transform frequency function given going hence Hilbert implies important impulse increasing inequality input integrable interval inverse Lebesgue Lesson limit linear means measurable method multiplications norm Note obtained operator orthogonal periodic poles polynomials possible problem Proof properties Proposition proves realizable relation Remark response result Riemann integral sampling satisfies sequence Show signal slowly solution space spectrum step sufficient Suppose Take tempered tends Theorem theory tion unique values vector wavelet wish write zero