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An Introduction to Basic Fourier Series

  • Book
  • © 2003

Overview

Authors:
  1. Sergei K. Suslov

    1. Department of Mathematics and Statistics, Arizona State University, Tempe, USA

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Part of the book series: Developments in Mathematics (DEVM, volume 9)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introduction

    • Sergei K. Suslov
    Pages 1-9

  3. Basic Exponential and Trigonometric Functions

    • Sergei K. Suslov
    Pages 11-46

  4. Addition Theorems

    • Sergei K. Suslov
    Pages 47-74

  5. Some Expansions and Integrals

    • Sergei K. Suslov
    Pages 75-102

  6. Introduction of Basic Fourier Series

    • Sergei K. Suslov
    Pages 103-136

  7. Investigation of Basic Fourier Series

    • Sergei K. Suslov
    Pages 137-184

  8. Completeness of Basic Trigonometric Systems

    • Sergei K. Suslov
    Pages 185-206

  9. Improved Asymptotics of Zeros

    • Sergei K. Suslov
    Pages 207-229

  10. Some Expansions in Basic Fourier Series

    • Sergei K. Suslov
    Pages 231-262

  11. Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function

    • Sergei K. Suslov
    Pages 263-292

  12. Numerical Investigation of Basic Fourier Series

    • Sergei K. Suslov
    Pages 293-321

  13. Suggestions for Further Work

    • Sergei K. Suslov
    Pages 323-326

  14. Back Matter

    Pages 327-371
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Keywords

About this book

It was with the publication of Norbert Wiener's book ''The Fourier In­ tegral and Certain of Its Applications" [165] in 1933 by Cambridge Univer­ sity Press that the mathematical community came to realize that there is an alternative approach to the study of c1assical Fourier Analysis, namely, through the theory of c1assical orthogonal polynomials. Little would he know at that time that this little idea of his would help usher in a new and exiting branch of c1assical analysis called q-Fourier Analysis. Attempts at finding q-analogs of Fourier and other related transforms were made by other authors, but it took the mathematical insight and instincts of none other then Richard Askey, the grand master of Special Functions and Orthogonal Polynomials, to see the natural connection between orthogonal polynomials and a systematic theory of q-Fourier Analysis. The paper that he wrote in 1993 with N. M. Atakishiyev and S. K Suslov, entitled "An Analog of the Fourier Transform for a q-Harmonic Oscillator" [13], was probably the first significant publication in this area. The Poisson k~rnel for the contin­ uous q-Hermite polynomials plays a role of the q-exponential function for the analog of the Fourier integral under considerationj see also [14] for an extension of the q-Fourier transform to the general case of Askey-Wilson polynomials. (Another important ingredient of the q-Fourier Analysis, that deserves thorough investigation, is the theory of q-Fourier series.

Authors and Affiliations

  • Department of Mathematics and Statistics, Arizona State University, Tempe, USA

    Sergei K. Suslov

Bibliographic Information

  • Book Title: An Introduction to Basic Fourier Series

  • Authors: Sergei K. Suslov

  • Series Title: Developments in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-3731-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 2003

  • Hardcover ISBN: 978-1-4020-1221-1Published: 31 March 2003

  • Softcover ISBN: 978-1-4419-5244-8Published: 07 December 2010

  • eBook ISBN: 978-1-4757-3731-8Published: 09 March 2013

  • Series ISSN: 1389-2177

  • Series E-ISSN: 2197-795X

  • Edition Number: 1

  • Number of Pages: XVI, 372

  • Topics: Special Functions, Fourier Analysis

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