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

By Sergei Suslov

It was once with the book of Norbert Wiener's e-book ''The Fourier In­ tegral and likely of Its functions" [165] in 1933 by means of Cambridge Univer­ sity Press that the mathematical group got here to gain that there's another method of the examine of c1assical Fourier research, particularly, during the thought of c1assical orthogonal polynomials. Little may he comprehend at the moment that this little inspiration of his might aid bring in 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 similar transforms have been made by way of different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of unique features 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. okay Suslov, entitled "An Analog of the Fourier rework for a q-Harmonic Oscillator" [13], was once most likely the 1st major ebook during this quarter. The Poisson k~rnel for the contin­ uous q-Hermite polynomials performs a task of the q-exponential functionality for the analog of the Fourier indispensable lower than considerationj see additionally [14] for an extension of the q-Fourier rework to the overall case of Askey-Wilson polynomials. (Another vital aspect of the q-Fourier research, that merits thorough research, is the idea of q-Fourier series.

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