An Introduction to Basic Fourier Series

by

Sergei K. Suslov

Department of Mathematics and Statistics
Arizona State University
Tempe, AZ 85287-1804
USA

Kluwer Academic Publishers
DORDRECHT  /  BOSTON  /  LONDON

DEVELOPMENTS IN MATHEMATICS, VOL. 9

ISBN 1-4020-1221-7

Dedicated to Dick Askey on his 70th birthday

This is an introductionary volume on a novel theory of basic Fourier series, a new interesting research area in classical analysis and q-series. This research utilizes approximation theory, orthogonal polynomials, analytic functions, and numerical methods to study the branch of q-special functions dealing with basic analogs of Fourier series and their applications. The present theory has interesting applications and connections to general orthogonal basic hypergeometric functions, a q-analog of zeta function, and, possibly, quantum groups and mathematical physics.

Audience
Researchers and graduate students interested in recent developments in q-special functions and their applications.

http://www.wkap.nl/

Contents

Foreword (by Mizan Rahman)
Preface
Chapter 1. Introduction
    1.1. Some Basic Exponential Functions
    1.2. Basic Fourier Series
    1.3. About This Book
    1.4. Exercises for Chapter 1
Chapter 2. Basic Exponential  and Trigonometric Functions
    2.1. Differential Equation for Harmonic Motion
    2.2. Difference Analog of Equation for Harmonic Motion
    2.3. Basic Exponential Functions
    2.4. Basic Trigonometric Functions
    2.5. q-Linear and Linear Grids
    2.6. Exercises for Chapter 2
Chapter 3. Addition Theorems
    3.1. Introduction
    3.2. First Proof of Addition Theorem: Analytic Functions
    3.3. Second Proof of Addition Theorem: Product Formula
    3.4. Third Proof of Addition Theorem: Difference Equation
    3.5. Another Addition Theorem
    3.6. Addition Theorems on q-Linear and Linear Grids
    3.7. Application: Continuous q-Hermite Polynomials
    3.8. Exercises for Chapter 3
Chapter 4. Some Expansions and Integrals
    4.1. Main Results
    4.2. Proofs of (4.1.1)
    4.3. Proofs of (4.1.3)
    4.4. Orthogonality Property
    4.5. Ismail and Zhang Formula
    4.6. q-Exponentials and Connection Coefficient Problems
    4.7. More Expansions and Integrals
    4.8. Second Proof of Ismail, Rahman and Zhang Formula
    4.9. Miscellaneous Results
    4.10. Exercises for Chapter 4
Chapter 5. Introduction of Basic Fourier Series
    5.1. Preliminaries
    5.2. Orthogonality Property for q-Trigonometric Functions
    5.3. Formal Limit q->1^-
    5.4. Some Properties of Zeros
    5.5. Evaluation of Some Constants
    5.6. Orthogonality Relations for q-Exponential Functions
    5.7. Basic Fourier Series
    5.8. Some Basic Trigonometric Identities
    5.9. Exercises for Chapter 5
Chapter 6. Investigation of Basic Fourier Series
    6.1. Uniform Bounds
    6.2. Completeness of Basic Trigonometric System
    6.3. Asymptotics of Zeros
    6.4. Pointwise Asymptotics of Basis
    6.5. Bilinear Generating Functions
    6.6. Methods of Summation of Basic Fourier Series
    6.7. Basic Trigonometric System and q-Legendre Polynomials
    6.8. Analytic Continuation of Basic Fourier Series
    6.9. Miscellaneous Results
    6.10. Exercises for Chapter  6
Chapter 7. Completeness of Basic Trigonometric System
    7.1. Completeness in L^2 and q-Lommel Polynomials
    7.2. Completeness in L^p: General Results
    7.3. Example: Some Infinite Products
    7.4. Example: Basic Sine and Cosine Functions
    7.5. Example: Jackson's q-Bessel Functions
    7.6. Exercises for Chapter 7
Chapter 8. Improved Asymptotics of Zeros
    8.1. Interpretation of Zeros and Preliminary Results
    8.2. Lagrange Inversion Formula
    8.3. Asymptotics of k'(w) and k''(w)
    8.4. Improved Asymptotics
    8.5. Alternative Forms of c_2(q)
    8.6. Monotonicity of c_1(q)
    8.7. Exercises for Chapter 8
Chapter 9. Some Expansions in Basic Fourier Series
    9.1. Expansions of Some Polynomials
    9.2. Basic Sine and Cosine Functions
    9.3. Basic Exponential Functions
    9.4. Basic Cosecant and Cotangent Functions
    9.5. Some Concequences of Parseval's Identity
    9.6. More Expansions
    9.7. Even More Expansions
    9.8. Miscellaneous Results
    9.9. Exercises for Chapter 9
Chapter 10. Basic Bernoulli and Euler Polynomials and Numbers and q -Zeta Function
    10.1. Bernoulli Polynomials, Numbers and Their q -Extensions
    10.2. Some Properties of q-Bernoulli Polynomials
    10.3. Extension of q-Bernoulli Polynomials
    10.4. Basic Euler Polynomials and Numbers
    10.5. Some Properties of q-Euler Polynomials
    10.6. Extensions of Riemann Zeta Function and Related Functions
    10.7. Analytic Continuation of q-Zeta Function
    10.8. Exercises for Chapter 10
Chapter 11. Numerical Investigation of Basic Fourier Series
    11.1. Eigenvalues
    11.2. Euler-Rayleigh Method
    11.3. Lower and Upper Bounds
    11.4. Eigenfunctions
    11.5. Some Examples of Basic Fourier Series and Related Sums
    11.6. Exercises for Chapter 11
Chapter 12. Suggestions for Further Work
Appendix A. Selected Summation and Transformation Formulas and Integrals
    A.1. Basic Hypergeometric Series
    A.2. Selected Summation Formulas
    A.3. Selected Transformation Formulas
    A.4. Some Basic Integrals
Appendix B. Some Theorems of Complex Analysis
    B.1. Entire Functions
    B.2. Proof of Lagrange's Inversion Formula
    B.3. Dirichlet Series
    B.4. Asymptotics
Appendix.C. Tables of Zeros of Basic Sine and Cosine Functions
Appendix D. Numerical Examples of Improved Asymptotics
Appendix E. Numerical Examples of Euler-Rayleigh Method
Appendix F. Numerical Examples of Lower and Upper Bounds
Bibliography
Index

Errata, updates of the References, etc., (as of December 3, 2003) ( dvi , ps , pdf )

Christian Reiher proved Conjecture 8.1 on page 223 and Conjecture 8.2 on page 226!  (reiher.dvi , reiher.ps , reiher.pdf )

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