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An Introduction to Basic Fourier Series (Developments in by Sergei Suslov

By Sergei Suslov

It used to be with the e-book of Norbert Wiener's ebook ''The Fourier In­ tegral and sure of Its purposes" [165] in 1933 through Cambridge Univer­ sity Press that the mathematical neighborhood got here to gain that there's an alternate method of the research of c1assical Fourier research, specifically, in the course of the idea of c1assical orthogonal polynomials. Little could he be aware of at the moment that this little proposal of his could support herald a brand new and exiting department of c1assical research referred to as q-Fourier research. makes an attempt at discovering q-analogs of Fourier and different comparable transforms have been made via different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of exact services and Orthogonal Polynomials, to determine the normal connection among orthogonal polynomials and a scientific concept of q-Fourier research. The paper that he wrote in 1993 with N. M. Atakishiyev and S. ok Suslov, entitled "An Analog of the Fourier rework for a q-Harmonic Oscillator" [13], was once most likely the 1st major e-book during this zone. The Poisson k~rnel for the contin­ uous q-Hermite polynomials performs a job of the q-exponential functionality for the analog of the Fourier critical lower than considerationj see additionally [14] for an extension of the q-Fourier rework to the overall case of Askey-Wilson polynomials. (Another vital factor of the q-Fourier research, that merits thorough research, is the idea of q-Fourier series.

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