Download An Introduction to Basic Fourier Series by Sergei Suslov PDF

By Sergei Suslov

It was once with the e-book of Norbert Wiener's ebook ''The Fourier In­ tegral and sure of Its purposes" [165] in 1933 via Cambridge Univer­ sity Press that the mathematical group got here to gain that there's another method of the research of c1assical Fourier research, specifically, in the course of the idea of c1assical orthogonal polynomials. Little might he recognize at the moment that this little thought 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 by way of different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of exact capabilities and Orthogonal Polynomials, to determine the normal connection among orthogonal polynomials and a scientific concept 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 remodel for a q-Harmonic Oscillator" [13], used to be 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 indispensable lower than considerationj see additionally [14] for an extension of the q-Fourier rework to the overall 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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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! J n=O I n. + eirp - i8 a/ß) ) n = exp(ax + ßy) by the binomial theorem. 16). Methods of solving of the difference equations of hypergeometrie type are discussed in [17], [139] and [140]j see also Exs.

BASIC EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS I_ Show that under the conditioIl8 lT (x) p (x) x k s-a,b = 0 with k 0,1,2, ... 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.

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. Let a he a root of the equation u (z) = O. Show that the differential equation of hypergeometric type has a power series solution of the form 00 y(z) = LCn(z-a)n, n=O where Cn+l c;: = if: dk (a) lim d m-too x kYm A + n (T' + (n -1)u" /2) (n + 1) (T (a) + nq' (a» , dk (z) = d x kY (z) for k = 0,1,2; (h) lim (A - Am) Cm (z - alm = 0; m-too withYm(z) = L:'=oCn(z-at ,Am = -mr'-m(m-1)u"/2.

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