By Sergei Suslov

It used to be with the e-book of Norbert Wiener's e-book ''The Fourier In­ tegral and sure of Its functions" [165] in 1933 via Cambridge Univer­ sity Press that the mathematical group got here to achieve that there's an alternate method of the examine of c1assical Fourier research, particularly, during the idea of c1assical orthogonal polynomials. Little might he understand at the moment that this little concept 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 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 detailed capabilities and Orthogonal Polynomials, to determine the traditional connection among orthogonal polynomials and a scientific idea 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 remodel for a q-Harmonic Oscillator" [13], was once most likely the 1st major book during this sector. 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 vital less than considerationj see additionally [14] for an extension of the q-Fourier rework to the final case of Askey-Wilson polynomials. (Another very important component of the q-Fourier research, that merits thorough research, is the speculation of q-Fourier series.

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