An Introduction to Basic Fourier Series

by Suslov, Sergei K.

• ISBN13: 9781402012211
• ISBN10: 1402012217
• Format: Hardcover
• Copyright: 2003-05-01
• Publisher: Kluwer Academic Pub

Summary

This is an introductory 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 its applications. This 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.

Table of Contents

Foreword	xi
Preface	xiii
Introduction	1	(10)
Some Basic Exponential Functions	1	(2)
Basic Fourier Series	3	(2)
About this Book	5	(1)
Exercises for Chapter 1	6	(5)
Basic Exponential and Trigonometric Functions	11	(36)
Differential Equation for Harmonic Motion	11	(2)
Difference Analog of Equation for Harmonic Motion	13	(3)
Basic Exponential Functions	16	(7)
Basic Trigonometric Functions	23	(5)
q-Linear and Linear Grids	28	(2)
Exercises for Chapter 2	30	(17)
Addition Theorems	47	(28)
Introduction	47	(1)
First Proof of Addition Theorem: Analytic Functions	47	(3)
Second Proof of Addition Theorem: Product Formula	50	(3)
Third Proof of Addition Theorem: Difference Equation	53	(5)
Another Addition Theorem	58	(3)
Addition Theorems on q-Linear and Linear Grids	61	(2)
Application: Continuous q-Hermite Polynomials	63	(3)
Exercises for Chapter 3	66	(9)
Some Expansions and Integrals	75	(28)
Main Results	75	(2)
Proofs of (4.1.1)	77	(1)
Proof of (4.1.3)	78	(2)
Orthogonality Property	80	(2)
Ismail and Zhang Formula	82	(1)
q-Exponentials and Connection Coefficient Problems	83	(7)
More Expansions and Integrals	90	(2)
Second Proof of Ismail, Rahman and Zhang Formula	92	(4)
Miscellaneous Results	96	(2)
Exercises for Chapter 4	98	(5)
Introduction of Basic Fourier Series	103	(34)
Preliminaries	103	(2)
Orthogonality Property for q-Trigonometric Functions	105	(6)
Formal Limit q → 1-	111	(1)
Some Properties of Zeros	112	(2)
Evaluation of Some Constants	114	(3)
Orthogonality Relations for q-Exponential Functions	117	(2)
Basic Fourier Series	119	(1)
Some Basic Trigonometric Identities	120	(2)
Exercises for Chapter 5	122	(15)
Investigation of Basic Fourier Series	137	(48)
Uniform Bounds	137	(3)
Completeness of Basic Trigonometric System	140	(6)
Asymptotics of Zeros	146	(9)
Pointwise Asymptotics of Basis	155	(2)
Bilinear Generating Function	157	(2)
Methods of Summation of Basic Fourier Series	159	(5)
Basic Trigonometric System and q-Legendre Polynomials	164	(5)
Examples	169	(2)
Analytic Continuation of Basic Fourier Series	171	(4)
Miscellaneous Results	175	(1)
Exercises for Chapter 6	176	(9)
Completeness of Basic Trigonometric Systems	185	(22)
Completeness in L2 and q-Lommel Polynomials	185	(7)
Completeness in LP: General Results	192	(7)
Example: Some Infinite Products	199	(2)
Example: Basic Sine and Cosine Functions	201	(1)
Example: Jackson's q-Bessel Functions	202	(1)
Exercises for Chapter 7	203	(4)
Improved Asymptotics of Zeros	207	(24)
Interpretation of Zeros and Preliminary Results	207	(3)
Lagrange Inversion Formula	210	(1)
Asymptotics of k' (w) and k'' (w)	211	(4)
Improved Asymptotics	215	(5)
Alternative Forms of c2 (q)	220	(3)
Monotonicity of c1 (q)	223	(3)
Exercises for Chapter 8	226	(5)
Some Expansions in Basic Fourier Series	231	(32)
Expansions of Some Polynomials	231	(7)
Basic Cosine and Sine Functions	238	(2)
Basic Exponential Function	240	(7)
Basic Cosecant and Cotangent Functions	247	(2)
Some Consequences of Parseval's Identity	249	(2)
More Expansions	251	(2)
Even More Expansions	253	(3)
Miscellaneous Results	256	(1)
Exercises for Chapter 9	257	(6)
Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function	263	(30)
Bernoulli Polynomials, Numbers and Their q-Extensions	263	(3)
Some Properties of q-Bernoulli Polynomials	266	(4)
Extension of q-Bernoulli Polynomials	270	(3)
Basic Euler Polynomials and Numbers	273	(2)
Some Properties of q-Euler Polynomials	275	(4)
Extensions of Riemann Zeta and Related Functions	279	(4)
Analytic Continuation of q-Zeta Function	283	(5)
Exercises for Chapter 10	288	(5)
Numerical Investigation of Basic Fourier Series	293	(30)
Eigenvalues	293	(5)
Euler-Rayleigh Method	298	(5)
Lower and Upper Bounds of Zeros	303	(6)
Eigenfunctions	309	(5)
Some Examples of Basic Fourier Series and Related Sums	314	(6)
Exercises for Chapter 11	320	(3)
Suggestions for Further Work	323	(4)
Appendix A. Selected Summation and Transformation Formulas and Integrals	327	(8)
A.1. Basic Hypergeometric Series	327	(1)
A.2. Selected Summation Formulas	328	(2)
A.3. Selected Transformation Formulas	330	(2)
A.4. Some Basic Integrals	332	(3)
Appendix B. Some Theorems of Complex Analysis	335	(10)
B.1. Entire Functions	335	(3)
B.2. Lagrange Inversion Formula	338	(2)
B.3. Dirichlet Series	340	(2)
B.4. Asymptotics	342	(3)
Appendix C. Tables of Zeros of Basic Sine and Cosine Functions	345	(4)
Appendix D. Numerical Examples of Improved Asymptotics	349	(2)
Appendix E. Numerical Examples of Euler-Rayleigh Method	351	(2)
Appendix F. Numerical Examples of Lower and Upper Bounds	353	(2)
Bibliography	355	(8)
Index	363