By Sergei Suslov

ISBN-10: 1441952446

ISBN-13: 9781441952448

ISBN-10: 1475737319

ISBN-13: 9781475737318

It used to be with the ebook of Norbert Wiener's publication ''The Fourier In tegral and likely of Its purposes" [165] in 1933 by way 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, in the course of the thought of c1assical orthogonal polynomials. Little may he recognize at the moment that this little notion of his could support 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 targeted features and Orthogonal Polynomials, to determine the traditional 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. okay Suslov, entitled "An Analog of the Fourier remodel for a q-Harmonic Oscillator" [13], was once most likely the 1st major booklet during this zone. 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 vital below considerationj see additionally [14] for an extension of the q-Fourier rework to the overall 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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Then show that ß (u (z) p(z) W (Y1, Y2» = (Al - A2) Y1 (z) Y2 (z) P (z) VX1 (z) , 40 2. BASIC EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS where Y1 = Y1 (Z) and Y2 = Y2 (Z) are solutions eorresponding to the eigenvalues Al and A2, respeetively, and Y1(Z) W (YI, Y2) Y2(Z) = VYI(Z) VY2(Z) VX(Z) VX(Z) Y2(Z) YI (Z - 1) - YI (Z) Y2(Z - 1) X(Z) - X(Z - 1) is a difference analog of the Wronskian. (22) Glassical orthogonal polynomials in a discrete variable. Let x (z) be a nonuniform lattice of the classical type eonsidered in Ex.

10); see Ex. 28. 4 ofthe next chapter. Eqs. ) q,q 00 (. 1), respectively. Eqs. 26) imply also the following representation . _ (_ aq l/4 ei8, _ aq l/4 e-i8;ql/2)00 cq (x, a) - (2. 5) from Appendix A. Eq. 32) shows that the function (qa 2; q2) cq (x; a) is an entire function in a and z when ei8 = qZ. 8). 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).

N. [Note: Among polynomial solutions of the differential equation of hypergeometric type are the Jacobi, Laguerre and Hermite polynomials [1], [10], [37], [113], [151]. ] (15) Integral representations. Let p (z) satisfy the Pearson equation [lT (z) P (z)]' = r (z) p (z) and let v be a root of the equation 1 A + vr' + 2"v (v -1) lT" = O. L = Vj (b) the contour C is chosen so that the equality lT"'(s)p(s) 181 =0 (s - z)"'+1 82 holds, where SI and s2 are the end points of the contour C. 6. EXERCISES FOR CHAPTER 2 37 (16) Power series method.

### An Introduction to Basic Fourier Series by Sergei Suslov

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