Fourier and Wavelet Analysis (Universitext)

this Book Is Intended As An Introduction To Classical Fourier Analysis, Fourier Series, And The Fourier Transform. The Topics Are Developed Slowly For The Reader Who Has Never Seen Them Before, With A Preference For Clarity Of Exposition In Stating And Proving Results. More Recent Developments, Such As The Discrete And Fast Fourier Transforms And Wavelets, Are Covered In The Last Two Chapters. The First Three, Short, Chapters Present Requisite Background Material, And These Could Be Read As A Short Course In Functional Analysis. The Text Includes Many Historical Notes To Place The Material In A Cultural And Mathematical Context; From The Fact That Jean Baptiste Joseph Fourier Was The Nineteenth, But Not The Last, Child In His Family To The Impact That Fourier Series Have Had On The Evolution Of The Concept Of The Integral. 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. "This book is intended as an introduction to classical Fourier analysis, Fourier series, and the Fourier transform. The topics are developed slowly for the reader who has never seen them before, with a preference for clarity of exposition in stating and proving results. More recent developments, such as the discrete and fast Fourier transforms and wavelets, are covered in the last two chapters. The first three, short, chapters present requisite background material, and these could be read as a short course in functional analysis. The text includes many historical notes to place the material in a cultural and mathematical context; from the fact that Jean Baptiste Joseph Fourier was the nineteenth, but not the last, child in his family to the impact that Fourier series have had on the evolution of the concept of the integral."--Jacket This comprehensive volume develops all of the standard features of Fourier analysis -- Fourier series, Fourier transform, Fourier sine and cosine transforms, and wavelets. The book's approach emphasizes the role of the "selector" functions, and is not embedded in the usual engineering context, which makes the material more accessible to a wider audience. While there are several publications on the various individual topics, none combine or even include all of the above. Metric and normed spaces -- Analysis -- Bases -- Fourier series -- The Fourier transform -- The discrete and fast Fourier transforms -- Wavelets. It is natural to think about distance between physical objects-people, say, or buildings or stars.