# Fourier and Wavelet Analysis

• George Bachman
• Lawrence Narici
• Edward Beckenstein
Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-ix
2. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 1-33
3. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 35-88
4. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 89-137
5. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 139-261
6. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 263-381
7. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 383-410
8. George Bachman, Lawrence Narici, Edward Beckenstein
Pages 411-487
9. Back Matter
Pages 489-505

### Introduction

globalized Fejer's theorem; he showed that the Fourier series for any f E Ld-7I", 7I"] converges (C, 1) to f (t) a.e. The desire to do this was part of the reason that Lebesgue invented his integral; the theorem mentioned above was one of the first uses he made of it (Sec. 4.18). Denjoy, with the same motivation, extended the integral even further. Concurrently, the emerging point of view that things could be decom­ posed into waves and then reconstituted infused not just mathematics but all of science. It is impossible to quantify the role that this perspective played in the development of the physics of the nineteenth and twentieth centuries, but it was certainly great. Imagine physics without it. We develop the standard features of Fourier analysis-Fourier series, Fourier transform, Fourier sine and cosine transforms. We do NOT do it in the most elegant way. Instead, we develop it for the reader who has never seen them before. We cover more recent developments such as the discrete and fast Fourier transforms and wavelets in Chapters 6 and 7. Our treatment of these topics is strictly introductory, for the novice. (Wavelets for idiots?) To do them properly, especially the applications, would take at least a whole book.

### Keywords

Fourier transform analysis convolution discrete Fourier transform fast Fourier transform fast Fourier transform (FFT) functional analysis

#### Authors and affiliations

• George Bachman
• 1
• Lawrence Narici
• 2
• Edward Beckenstein
• 3
1. 1.Emeritus of MathematicsPolytechnic UniversityBrooklynUSA
2. 2.Department of Mathematics and Computer ScienceSt. John’s UniversityJamaicaUSA
3. 3.Science DivisionSt. John’s UniversityStaten IslandUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-0505-0
• Copyright Information Springer-Verlag New York, Inc. 2000
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-6793-5
• Online ISBN 978-1-4612-0505-0
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675
• Buy this book on publisher's site