About the Book
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.
Table of Contents:
1 Metrie and Normed Spaces.- 1.1 Metrie Spaces.- 1.2 Normed Spaces.- 1.3 Inner Product Spaces.- 1.4 Orthogonality.- 1.5 Linear Isometry.- 1.6 Holder and Minkowski Inequalities; Lpand lpSpaces..- 2 Analysis.- 2.1 Balls.- 2.2 Convergence and Continuity.- 2.3 Bounded Sets.- 2.4 Closure and Closed Sets.- 2.5 Open Sets.- 2.6 Completeness.- 2.7 Uniform Continuity.- 2.8 Compactness.- 2.9 Equivalent Norms.- 2.10 Direct Sums.- 3 Bases.- 3.1 Best Approximation.- 3.2 Orthogonal Complements and the Projection Theorem.- 3.3 Orthonormal Sequences.- 3.4 Orthonormal Bases.- 3.5 The Haar Basis.- 3.6 Unconditional Convergence.- 3.7 Orthogonal Direct Sums.- 3.8 Continuous Linear Maps.- 3.9 Dual Spaces.- 3.10 Adjoints.- 4 Fourier Series.- 4.1 Warmup.- 4.2 Fourier Sine Series and Cosine Series.- 4.3 Smoothness.- 4.4 The Riemann-Lebesgue Lemma.- 4.5 The Dirichlet and Fourier Kernels.- 4.6 Point wise Convergence of Fourier Series.- 4.7 Uniform Convergence.- 4.8 The Gibbs Phenomenon.- 4.9 — Divergent FourierSeries.- 4.10 Termwise Integration.- 4.11 Trigonometric vs. Fourier Series.- 4.12 Termwise Differentiation.- 4.13 Dido’s Dilemma.- 4.14 Other Kinds of Summability.- 4.15 Fejer Theory.- 4.16 The Smoothing Effect of (C, 1) Summation.- 4.17 Weierstrass’s Approximation Theorem.- 4.18 Lebesgue’s Pointwise Convergence Theorem.- 4.19 Higher Dimensions.- 4.20 Convergence of Multiple Series.- 5 The Fourier Transform.- 5.1 The Finite Fourier Transform.- 5.2 Convolution on T.- 5.3 The Exponential Form of Lebesgue’s Theorem.- 5.4 Motivation and Definition.- 5.5 Basics/Examplesv.- 5.6 The Fourier Transform and Residues.- 5.7 The Fourier Map.- 5.8 Convolution on R.- 5.9 Inversion, Exponential Form.- 5.10 Inversion, Trigonometric Form.- 5.11 (C, 1) Summability for Integrals.- 5.12 The Fejer-Lebesgue Inversion Theorem.- 5.13 Convergence Assistance.- 5.14 Approximate Identity.- 5.15 Transforms of Derivatives and Integrals.- 5.16 Fourier Sine and Cosine Transforms.- 5.17 Parseval’s Identities.- 5.18 TheL2Theory.- 5.19 The Plancherel Theorem.- 5.20 Point wise Inversion and Summability.- 5.21 — Sampling Theorem.- 5.22 The Mellin Transform.- 5.23 Variations.- 6 The Discrete and Fast Fourier Transforms.- 6.1 The Discrete Fourier Transform.- 6.2 The Inversion Theorem for the DFT.- 6.3 Cyclic Convolution.- 6.4 Fast Fourier Transform for N=2k.- 6.5 The Fast Fourier Transform for N=RC.- 7 Wavelets.- 7.1 Orthonormal Basis from One Function.- 7.2 Multiresolution Analysis.- 7.3 Mother Wavelets Yield Wavelet Bases.- 7.4 From MRA to Mother Wavelet.- 7.5 Construction of — Scaling Function with Compact Support.- 7.6 Shannon Wavelets.- 7.7 Riesz Bases and MRAs.- 7.8 Franklin Wavelets.- 7.9 Frames.- 7.10 Splines.- 7.11 The Continuous Wavelet Transform.
Review :
From the reviews: "This excellent book is intended as an introduction to classical Fourier analysis, Fourier series, Fourier transforms and wavelets, for students in mathematics, physics, and engineering. The text includes many historical notes to place the material in a cultural and mathematical context. The topics are developed slowly for the reader who has neverr seen them before, with a preference for clarity of exposition in stating and proving results." EMS Newsletter, Issue 39, March 2001 "The book under review is intended primarily as an introduction to classical Fourier analysis, Fourier series and Fourier transform. ! The discussion is thorough and shows the material at a leisurely pace. There are many exercises that expand on the material, followed by hints or answers. ! this book is a reader-friendly, gentle introduction to the theory of Fourier analysis. I recommend this book to graduate or postgraduate mathematics and physics students, engineers, computer scientists and everybody who want to learn Fourier Analysis ! ." (Zoltan Nemeth, Acta Scientiarum Mathematicarum, Vol. 71, 2005) "This book is a self-contained treatise on Fourier analysis and wavelet theory. ! is a nice textbook enriched by a lot of historical notes and remarks ! . Definitions and results are illustrated by examples. The text, together with the exercises at the end of each section, covers important results. Moreover, most of the exercises are followed by hints on how to solve them, answers or references. Both the style of writing and the contents are quite pleasant and easy to follow ! ." (Kathi Karima Selig, Mathematical Reviews, Issue 2001 a) "I enjoyed reading it and learnt quite a lot from it. ! The author's general intention was ! to write a book that would present the prerequisites for wavelet theory ! . There are also exercises by the way. ! and are furnished with 'hints' that are usually full solutions. ! It is informative, interestingly and clearly written with intelligent comments and pleasing explanations, a delight to read." (Christopher Atkin, The New Zealand Mathematical Society Newsletter, Issue 84, 2002) "Fourier and Wavelet Analysis is primarily an introduction to the theory of Fourier series and Fourier transforms ! . There are many sets of exercises that expand on the material in the text, each followed by a corresponding set of hints and/or answers. Accordingly this book is well suited for self-study." (Gerald B. Folland, SIAM Review, Vol. 43 (1), 2000) "The presentation of Fourier Analysis given in this book follows historical development of the subject. ! The material is given in the classical style. It is very detailed and self-contained ! . Each chapter comes with a collection of exercises." (H. G. Feichtinger, Monatshefte fur Mathematik, Vol. 131 (4), 2001) "The text is concerned with the classical theory of Fourier series and transforms ! . there are plenty of worked examples and exercises, with hints and solutions. There is also an extensive list of references for readers who wish to investigate further. This is a serious and scholarly work which should be in the library of every mathematics department." (Gerry Leversha, The Mathematical Gazette, Vol. 85 (502), 2001) "This is a good introduction to classical Fourier analysis ! . More recent developments such as the discrete and fast Fourier transforms are also covered. ! The book includes many historical notes and useful background material from functional analysis." (B. Rubin, Zentralblatt MATH, Vol. 948, 2000) "A specific feature of the book under review is that it combines a simple, clear and intuitive exposition of the results, together with a very rigorous mathematical treatment. ! The volume is an excellent text book for graduates and professionals in mathematics, engineering and physical sciences. ! . The text includes many historical notes to place the material in a cultural and mathematical context. The main merit of the book is to provide beginners in Fourier analysis with exactly what they are looking for." (B. Kirstein, Zeitschrift fur Analysis und ihre Anwendungen, Vol. 19 (3), 2000) "An approachable text for the advanced undergraduate; but perhaps more suitable as a recommended text for early postgraduate study. The text is fluently written with many historical details that puts Fourier's work into the context of his contemporaries and those who tread his footsteps. Fourier's series, transforms and wavelet theory are covered in full ! ." (ASLIB Book Guide, Vol. 65 (5), 2000)
