Ebook: An Introduction to Basic Fourier Series

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.

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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.

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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.

Content:
Front Matter....Pages i-xv
Introduction....Pages 1-9
Basic Exponential and Trigonometric Functions....Pages 11-46
Addition Theorems....Pages 47-74
Some Expansions and Integrals....Pages 75-102
Introduction of Basic Fourier Series....Pages 103-136
Investigation of Basic Fourier Series....Pages 137-184
Completeness of Basic Trigonometric Systems....Pages 185-206
Improved Asymptotics of Zeros....Pages 207-229
Some Expansions in Basic Fourier Series....Pages 231-262
Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function....Pages 263-292
Numerical Investigation of Basic Fourier Series....Pages 293-321
Suggestions for Further Work....Pages 323-326
Back Matter....Pages 327-371

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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.

Content:
Front Matter....Pages i-xv
Introduction....Pages 1-9
Basic Exponential and Trigonometric Functions....Pages 11-46
Addition Theorems....Pages 47-74
Some Expansions and Integrals....Pages 75-102
Introduction of Basic Fourier Series....Pages 103-136
Investigation of Basic Fourier Series....Pages 137-184
Completeness of Basic Trigonometric Systems....Pages 185-206
Improved Asymptotics of Zeros....Pages 207-229
Some Expansions in Basic Fourier Series....Pages 231-262
Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function....Pages 263-292
Numerical Investigation of Basic Fourier Series....Pages 293-321
Suggestions for Further Work....Pages 323-326
Back Matter....Pages 327-371
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