## The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet ApproximationsThis book represents the first attempt at a unified picture for the pres ence of the Gibbs (or Gibbs-Wilbraham) phenomenon in applications, its analysis and the different methods of filtering it out. The analysis and filtering cover the familiar Gibbs phenomenon in Fourier series and integral representations of functions with jump discontinuities. In ad dition it will include other representations, such as general orthogonal series expansions, general integral transforms, splines approximation, and continuous as well as discrete wavelet approximations. The mate rial in this book is presented in a manner accessible to upperclassmen and graduate students in science and engineering, as well as researchers who may face the Gibbs phenomenon in the varied applications that in volve the Fourier and the other approximations of functions with jump discontinuities. Those with more advanced backgrounds in analysis will find basic material, results, and motivations from which they can begin to develop deeper and more general results. We must emphasize that the aim of this book (the first on the sUbject): to satisfy such a diverse audience, is quite difficult. In particular, our detailed derivations and their illustrations for an introductory book may very well sound repeti tive to the experts in the field who are expecting a research monograph. To answer the concern of the researchers, we can only hope that this book will prove helpful as a basic reference for their research papers. |

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### Contents

IV | 1 |

V | 3 |

VI | 12 |

VIII | 16 |

IX | 26 |

X | 28 |

XI | 31 |

XII | 34 |

XXXIX | 137 |

XL | 140 |

XLI | 148 |

XLII | 150 |

XLIII | 155 |

XLIV | 156 |

XLVI | 157 |

XLVII | 159 |

XIII | 37 |

XIV | 38 |

XV | 40 |

XVI | 43 |

XVII | 44 |

XVIII | 45 |

XIX | 55 |

XX | 56 |

XXI | 57 |

XXII | 67 |

XXIII | 73 |

XXIV | 78 |

XXV | 80 |

XXVI | 83 |

XXVII | 100 |

XXVIII | 107 |

XXIX | 109 |

XXX | 110 |

XXXI | 112 |

XXXII | 119 |

XXXIII | 122 |

XXXV | 125 |

XXXVI | 127 |

XXXVII | 130 |

XXXVIII | 131 |

### Common terms and phrases

a-factor asymptotic basic Bessel functions coefficients computations condition continuous wavelet transform convergence cosine Courtesy Daubechies decaying Dirichlet kernel discrete wavelet discuss end points example exponential extremas Fejer averaging Figure finite Fourier analysis Fourier integral representation Fourier series Fourier transform Fourier-Bessel series function f(x functions with jump gate function Gegenbauer Gibbs overshoot Gibbs phe Gibbs phenomenon Gottlieb Hewitt and Hewitt interval ISBN jump discontinuity Karanikas Lanczos limit Math maximum overshoot method Mexican hat wavelet nomenon Nth partial sum orthogonal expansion oscillations overshoots and undershoots piecewise Pinsky Poisson wavelet polynomial expansion reducing the Gibbs result sawtooth function scaling function Section series expansion sgn(t shown in Fig signum function sine integral smooth spline square integrable square wave square wave function summability Tchebychev polynomials theorem tion transform representation trigonometric functions trigonometric polynomial truncated Fourier integral truncated Fourier series unit step function wavelet analysis wavelet ip(t Wilbraham's zero

### Popular passages

Page 288 - RG Cooke, Gibbs phenomenon in Fourier-Bessel series and integrals, Proc. London Math. Soc. (2) 27 (1928), 171.

Page 293 - L. Fejér, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Funktion beziehen,

Page 288 - Error Analysis in Application of Generalization of the Sampling Theorem - chapter 7 in Advanced Topics in Shannon Sampling and Interpolation Theory, RJ Marks II, editor, SpringerVerlag, 1992,