By Sergei Suslov
It used to be with the booklet of Norbert Wiener's publication ''The Fourier In tegral and sure of Its functions"  in 1933 via Cambridge Univer sity Press that the mathematical group got here to achieve that there's another method of the research of c1assical Fourier research, particularly, in the course of the conception of c1assical orthogonal polynomials. Little might he understand at the moment that this little thought of his may aid 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 distinctive features and Orthogonal Polynomials, to determine the ordinary connection among orthogonal polynomials and a scientific conception 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" , used to be most likely the 1st major ebook 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 indispensable lower than considerationj see additionally  for an extension of the q-Fourier remodel to the overall case of Askey-Wilson polynomials. (Another vital component of the q-Fourier research, that merits thorough research, is the idea of q-Fourier series.
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Extra resources for An Introduction to Basic Fourier Series
Cq (x, y; a) = c q (x; a) c q (y; a) . 3) has attracted some attention and different proofs of this relation were given in , , and . Floreanini, LeTourneux, and Vinet  have used group theoretical methods. Ismail and Stanton gave two different proofs on the basis of the eonnection coefficients for the qHermite polynomials  and yet another proof in . See also  for the original proofby direet series manipulations and -. In this chapter we shall diseuss several different proofs of this result and its extension.
4) Verify the following identities 1 a (I-')'Y (I-') ="2 'Y(21-') , + (a 2 - 1) 'Y2 (I-') = a (21-'), a (I-' + 11) a (21-') - (a 2 - 1) 'Y (I-' + 11) 'Y (21-') = a (I-' - 11), a (I-' + 11) 'Y (21-') - 'Y (I-' + 11) a (21-') = 'Y (I-' - 11), a (-I-') = a (I-') , 'Y ( -I-') = -'Y (I-'), a (I-' - 1) 'Y (I-') - a (11 - 1) 'Y (11) = a (I-' + 11 - 1) 'Y (I-' - 11) , "Y(I-' - 1) 'Y (I-') - 'Y (11 - 1) 'Y (11) = 'Y (I-' + 11 - 1) 'Y (I-' - 11) , a 2 (I-') where the functions a (I-') and 'Y (I-') are defined in the previous exercise.
4. 4. Basie Trigonometrie Functions The basic cosine Cq (x, 1/j w) and basic sine Sq (x,1/jw) functions can be introduced by the following analog of the Eu1er formula Eq (x, 1/j iw) = Cq (x, 1/j w) + iSq (x, 1/j w). 3) -qw jq 00 -q _qeiO+irp,_qeiB-irp,_qeirp-iB,_qe-iB-irp 2) ( x 4c,oa _q, q3/2, _q3/2 j q,-w . 1), + 1/) and sinw (x + 1/), respectively. 7) are defined here for An analytic continuation of these functions in a Iarger domain was discussed in , , , and . For example, Iwi < 1 only.