An Introduction to Basic Fourier Series,9781402012211

An Introduction to Basic Fourier Series

by
ISBN13: 9781402012211
ISBN10: 1402012217
Format: Hardcover
Pub. Date: 2003-05-01
Publisher(s): Kluwer Academic Pub
List Price: $262.49

Rent Textbook

Select for Price
Add to Cart
There was a problem. Please try again later.

Rent Digital

Rent Digital Options
Online:30 Days access
Downloadable:30 Days
$57.24
Online:60 Days access
Downloadable:60 Days
$76.32
Online:90 Days access
Downloadable:90 Days
$95.40
Online:120 Days access
Downloadable:120 Days
$114.48
Online:180 Days access
Downloadable:180 Days
$124.02
Online:1825 Days access
Downloadable:Lifetime Access
$190.80
*To support the delivery of the digital material to you, a digital delivery fee of $3.99 will be charged on each digital item.
$124.02*
Add to Cart

New Textbook

We're Sorry
Sold Out

Used Textbook

We're Sorry
Sold Out

Buy from our Marketplace starting at $175.65

How Marketplace Works:

  • This item is offered by an independent seller and not shipped from our warehouse
  • Item details like edition and cover design may differ from our description; see seller's comments before ordering.
  • Sellers much confirm and ship within two business days; otherwise, the order will be cancelled and refunded.
  • Marketplace purchases cannot be returned to eCampus.com. Contact the seller directly for inquiries; if no response within two days, contact customer service.
  • Additional shipping costs apply to Marketplace purchases. Review shipping costs at checkout.

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

An electronic version of this book is available through VitalSource.

This book is viewable on PC, Mac, iPhone, iPad, iPod Touch, and most smartphones.

By purchasing, you will be able to view this book online, as well as download it, for the chosen number of days.

Digital License

You are licensing a digital product for a set duration. Durations are set forth in the product description, with "Lifetime" typically meaning five (5) years of online access and permanent download to a supported device. All licenses are non-transferable.

More details can be found here.

A downloadable version of this book is available through the eCampus Reader or compatible Adobe readers.

Applications are available on iOS, Android, PC, Mac, and Windows Mobile platforms.

Please view the compatibility matrix prior to purchase.