Algebraic numbers and Fourier analysis by Raphael Salem PDF

By Raphael Salem

ISBN-10: 0669192732

ISBN-13: 9780669192735

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Comptes Rendus de l'Acad. des Sciences (Paris), 92 (1881), 448-450. 12. G. H. Hardy and E. M. , Oxford University Press, Oxford, 1960. 13. K. D. thesis, The Pennsylvania State University, Nov. 1979. 14. L. 3. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966. 15. 3. 3. Sylvester, A constructive theory of partitions, arranged in three acts, an interact, and an exodion, Amer. , 5 (1882), 251-330 and 6 (1884), 334-336 (or pp. 1-83 of Collected Mathematical Papers of Games Joseph Sylvester, Vol.

11). 11). 11). 11) extends with rates of convergence 2(M+1) and . 2M+1, respectively, as required. For n^ 0 < N < 3, which has one expansion with rate of convergence The expansions of of we shall give an infinite family of expansions of m. [ ] = — "' v (yq) ' x n -[ ] = (xq)^. 10). 12) 1,y (yq). 10) of nn with rates of convergence respectively. 14) •1 E a > ^ ^ab 2*1 = <-) (~) 3 oo b °° 1 (c) (£-) 0 and 1, ab • oo ab ° which is known as the q-analog of Gauss' theorem. (in our more general sense) can be summed.

2 n n +n ( y yq ) q n + 1 ( a ) n GENERALIZED HYPERGEOMETRIC SERIES -J (1 2 0 * 2 "yq) 1 (1-yq) d + 27 *s + ... d-yq 2 )(1-q) / vn n + JXILH n ^ n (yq 2 ) n (q) r + ... 3) follows easily. We now formalize this process. Let II be or one of our infinite products n N , N >_ 0. 4) I t n=1 n = 1 + 1 or (xq) (yq). 6) n. b To this purpose we decompose each term n tn = a n + bn. We then rearrange our series so that the "a part" term. 1 + 1 of each term is grouped with the When lim a = 0 , n >» n is grouped with "b part" a,.

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Algebraic numbers and Fourier analysis by Raphael Salem


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