New PDF release: An Introduction to Basic Fourier Series

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By Sergei Suslov

ISBN-10: 1441952446

ISBN-13: 9781441952448

ISBN-10: 1475737319

ISBN-13: 9781475737318

It was once with the e-book of Norbert Wiener's booklet ''The Fourier In­ tegral and likely of Its functions" [165] in 1933 through Cambridge Univer­ sity Press that the mathematical group got here to achieve that there's an alternate method of the learn of c1assical Fourier research, particularly, in the course of the idea of c1assical orthogonal polynomials. Little may he recognize at the moment that this little suggestion of his may aid herald 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 similar transforms have been made through different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of specific capabilities and Orthogonal Polynomials, to determine the ordinary connection among orthogonal polynomials and a scientific thought 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], used to be most likely the 1st major book 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 essential below considerationj see additionally [14] for an extension of the q-Fourier rework to the final case of Askey-Wilson polynomials. (Another vital factor of the q-Fourier research, that merits thorough research, is the idea of q-Fourier series.

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Example text

1. The parallelogram of the double periodicity in the complex z-plane. 3) in the open disk lai< 8. 3) in the parameter a to the entire complex a-plane completes the proof. The following q-addition formulas for the basie trigonometrie functions have been established in [140]. 2. Cq (X,YiW) = Cq (XiW) Cq (YiW) - Bq (XiW) Bq (Yi W) , Bq (x, y;w) = Bq (x;w) Cq (y;w) + Cq (x;w) Bq (y;w) . PROOF. 1). 10) 3. 19) 2 eos2 WX, 2 sin2 WX, respectively. 12) if the Cq (Xj w) and Bq (Xj w) are real valued funetionsj in this ease Eq.

2) qa ,q with 1] = (q1/4 + q-1/4) /2. 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!

The polynomial solutions Yn (x) and Ym (x) are orthogonal on a contour 0 in the complex z-plane: [Yn(x(z)) Ym(x(z)) p(z)Vxdz) dz=O, mi=n. [Note: In many cases a special choice of the contour 0 in this relation gives rise to the continuous orthogonality property for a realvalued system of polynomials. The most general case corresponds to the Askey-Wilson polynomials [15], [18]; see other examples in [9], [10], [16], [46], [86], [111], [113]. ] (23) Show that for the nonuniform lattices of the classical type v(n) 1 (z) = t (_l)n-k k=O 'Y (n)!

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