# Fourier-Stieltjes series

## Fourier-Stieltjes series

[‚fu̇r·ē‚ā ′stēl·yes ‚sir·ēz]
(mathematics)
For a function ƒ(x) of bounded variation on the interval [0,2π], the series from n = 0 to infinity of cn exp (inx), where cn is 1/2π times the integral from x = 0 to x = 2π of exp (-inx) d ƒ(x).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
Given a symmetric stable process X(t) with the index [alpha] where 1 < [alpha] < 2 and f [member of] [L.sup.P] [0, 2[pi]], P [greater than or equal to] [alpha], the question of the convergence of associated random Fourier-Stieltjes series
For f [member of] [L.sup.2] [0, 2[pi]] and X is the standard Brownian motion process it is not hard to show that the random Fourier-Stieltjes series converge almost surely to the stochastic integral (4), as An are independent.
For a function f of monotonic type and of Class [[LAMBDA].sub.[beta]] with [beta] [member of] (0,1) and a symmetric stable process X of index [alpha] [member of] (1,2) with [alpha][beta] > 1 , the random Fourier-Stieltjes series [[infinity].summation over (n=-[infinity])] [a.sub.n][A.sub.n][e.sup.int] Converge almost surely to the stochastic integral
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