An Introduction to Basic Fourier Series

  • Sergei K. Suslov

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

Table of contents

  1. Front Matter
    Pages i-xv
  2. Sergei K. Suslov
    Pages 1-9
  3. Sergei K. Suslov
    Pages 11-46
  4. Sergei K. Suslov
    Pages 47-74
  5. Sergei K. Suslov
    Pages 75-102
  6. Sergei K. Suslov
    Pages 103-136
  7. Sergei K. Suslov
    Pages 137-184
  8. Sergei K. Suslov
    Pages 185-206
  9. Sergei K. Suslov
    Pages 207-229
  10. Sergei K. Suslov
    Pages 231-262
  11. Sergei K. Suslov
    Pages 293-321
  12. Sergei K. Suslov
    Pages 323-326
  13. Back Matter
    Pages 327-371

About this book

Introduction

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.

Keywords

Complex analysis Hypergeometric function Mathematica Volume analytic function approximation approximation theory form function mathematical physics numerical method special function theorem transformation zeta function

Authors and affiliations

  • Sergei K. Suslov
    • 1
  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-3731-8
  • Copyright Information Springer-Verlag US 2003
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-5244-8
  • Online ISBN 978-1-4757-3731-8
  • Series Print ISSN 1389-2177
  • About this book