Read e-book online An Introduction to Basic Fourier Series PDF

By Sergei Suslov

ISBN-10: 1441952446

ISBN-13: 9781441952448

ISBN-10: 1475737319

ISBN-13: 9781475737318

It was once with the ebook of Norbert Wiener's e-book ''The Fourier In­ tegral and likely of Its functions" [165] in 1933 by way of Cambridge Univer­ sity Press that the mathematical group got here to achieve that there's another method of the research of c1assical Fourier research, specifically, in the course of the concept of c1assical orthogonal polynomials. Little might he be aware of at the moment that this little suggestion of his may aid bring in a brand new and exiting department of c1assical research known as q-Fourier research. makes an attempt at discovering q-analogs of Fourier and different comparable transforms have been made by means of different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of precise capabilities and Orthogonal Polynomials, to work out 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. okay Suslov, entitled "An Analog of the Fourier remodel for a q-Harmonic Oscillator" [13], used to be most likely the 1st major book during this quarter. 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 crucial less than considerationj see additionally [14] for an extension of the q-Fourier rework to the final case of Askey-Wilson polynomials. (Another vital component of the q-Fourier research, that merits thorough research, is the speculation of q-Fourier series.

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3). 3. 18). 15) with x (z) = (cf + q-Z) /2 = coso, qZ = ei8 ; see Ex. 11. 11). 5, Eq. 1). 1). 18) (a2/ß2;q)kqk2/4 (ßei'P)k (q; q)k . xL00 qn(n-2k)/4 ßne-m'P (q; q)n n=O X ( _q(l- n+k)/2 ei8+ i'P a/ ß, _q(l-n+k)/2ei'P-i8 a/ ß; q) n . 1) onee again. The seeond sum ean be redueed to the sum of two 4CP3 series similar to those in 20 2. 17) is an analog of exp (ax + ßy). im Eq (x, Yj (1 - q) a/2, (1 - q) ß/2) q-tl- = f: (ß/~)n n=O n. e- inrp (1+e i8 +irp a/ß)n (1+e irp - i8a/ß)n n (-n)k ( x {; ~ =L (ß/2)n n!

I) ' and 0 is a contour in the complex z-plane, if: 42 2. BASIC EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS (a) p (z) and Pv (z) satisfy the Pearson type equations (z) P (z)] = 1" (z) P (z) VX1 (z) , [0" (z) Pv (z)] = 1"v (z) Pv (z) VXv +1 (z), ß ß [0" and 11 is a root of the equation Av + Cl( (11) l' (11) T' - l' (11 - 1) l' (11) 0:" /2 (b) the generalized powers [x (8) - (8) - [xv (z)] Xv = [Xv (8) = [Xv (8) [X II -1 (z)](~) Xv (z)](~+l) , X II -1 = [XII_~ (8 + JL) (8) - (8) - Xv (8 + 1) - = [XII [xv Xv Xv [X v -1 X = 0; (z)](~) have the properties (z -1)](~) (8) - Xv (z - JL)] (z)](~) [XII_~ (8) - XII_~ (z)] Xv-~ (z)] [X II -1 (8) - Xv (z)](~) (z)](M1) , (c) the difference-differentiation formula V'Pv~ (z) = l' (JL + 1) 'Pli, ~+1 (z) holds for JL = 11 and JL = 11 - 1; Vxv_~ (z) (d) the equations 0" [X II -1 (8) - (8) Pv (8) XII-l (z + 1)](11+1) b =0 a and { ß ( 0" (8) Pv (8) ) d8 = 0 Je [X II -1 (8) - XII-l (z + 1)](11+1) are satisfied in the cases of the sum and the integral, respectively.

11) ( -CIC2 (1 - q)2 ql/2 Alu; q2) 00 (-Cl C2 (1 - q)2 q-I/2 Alu; q2) 00 IC C We have used the q-binomial theorem when I 2 (1- q)2 q-I/2Alul < 1. 7). 8). 8) are, in fact, equivalent. 2. 27). 8) are well-defined when I 2 (1- q)2 q-I/2Aju l < 1. But the analytic continuation of the 2'PIS on the right side in to the larger domain can be easily given [46]. 8) one gets _l-'q1+ 2Z , _ql-2z I I-' 2'PI ( ; q ,( q (_l-'q1+2Z(, _ql-2Z(j1-'; q2) 00 PROOF.

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