| 000 | 00379nam a2200133Ia 4500 | ||
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| 999 |
_c177080 _d177080 |
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| 020 | _a0387008365 (acid-free paper) | ||
| 040 | _cCUS | ||
| 082 |
_a515.2433 _bVRE/F |
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| 100 | _aVretblad.,Anders | ||
| 245 | 0 |
_aFourier analysis and its applications/ _cAnders Vretblad. |
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| 260 |
_aNew York :: _bSpringer, _cc2003. |
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| 300 | _axi, 269 p. : | ||
| 505 | _a 1 Introduction 1 1.1 The classical partial differential equations 1 1.2 Well-posed problems 3 1.3 The one-dimensional wave equation 5 1.4 Fourier's method 9 2 Preparations 15 2.1 Complex exponentials 15 2.2 Complex-valued functions of a real variable 17 2.3 Cesaro summation of series 20 2.4 Positive summation kernels 22 2.5 The Riemann-Lebesgue lemma 25 2.6 *Some simple distributions 27 2.7 ^Computing with S 32 3 Laplace and Z transforms 39 3.1 The Laplace transform 39 3.2 Operations 42 3.3 Applications to differential equations 47 3.4 Convolution , 53 3.5 *Laplace transforms of distributions 57 3.6 The Z transform 60 X Contents 3.7 Applications in control theory 57 Summary of Chapter 3 70 4 Fourier series 73 4.1 Definitions 73 4.2 Dirichlet's and Fejer's kernels; uniqueness 80 4.3 Differentiable functions 34 4.4 Point wise convergence 35 4.5 Formulae for other periods qq 4.6 Some worked examples 91 4.7 The Gibbs phenomenon 93 4.8 *Fourier series for distributions 96 Summary of Chapter 4 5 Theory 105 5.1 Linear spaces over the complex numbers 105 5.2 Orthogonal projections HO 5.3 Some examples H4 5.4 The Fourier system is complete II9 5.5 Legendre polynomials I23 5.6 Other classical orthogonal polynomials 127 Summary of Chapter 5 130 6 Separation of variables 137 6.1 The solution of Fourier's problem I37 6.2 Variations on Fourier's theme I39 6.3 The Dirichlet problem in the unit disk I43 6.4 Sturm—Liouville problems 133 6.5 Some singular Sturm—Liouville problems I59 Summary of Chapter 6 IgO 7 Fourier transforms 13^ 7.1 Introduction Igg 7.2 Definition of the Fourier transform 166 7.3 Properties Igg 7.4 The inversion theorem 171 7.5 The convolution theorem 17g 7.6 Plancherel's formula IgQ 7.7 Application 1 Ig2 7.8 Application ^ 185 7.9 Application 3: The sampling theorem I37 7.10 Connection with the Laplace transform I33 7.11 *Distributions and Fourier transforms 19q Summary of Chapter 7 ! . . . ! 192 ' Contents xi 8 Distributions 197 8.1 History 197 8.2 Fuzzy points - test functions 200 8.3 Distributions 203 8.4 Properties 206 8.5 Fourier transformation 213 8.6 Convolution 218 8.7 Periodic distributions and Fourier series 220 8.8 Fundamental solutions 221 8.9 Back to the starting point 223 Summary of Chapter 8 224 9 Multi-dimensional Fourier analysis 227 9.1 Rearranging series 227 9.2 Double series 230 9.3 Multi-dimensional Fourier series 233 9.4 Multi-dimensional Fourier transforms 236 Appendices A The ubiquitous convolution 239 B The discrete Fourier transform 243 C Formulae 247 C.l Laplace transforms 247 C.2 Z transforms 250 C.3 Fourier series 251 C.4 Fourier transforms 252 C.5 Orthogonal polynomials 254 D Answers to selected exercises 257 E Literature 265 Index 267 | ||
| 942 | _cWB16 | ||