On the arithmetic mean of Fourier-Stieltjes coefficients

Abstract

Let {a"}^L0 be the cosine Fourier-Stieltjescoefficients of the Bore! measure /i and {a0, (a, +-• •+û«)/«i"=i = {(7a)"}^=0 be the sequence of their arithmetic means. Then ]>,íí=o (Tá)n cos nx is a Fourier-Stieltjes series. Moreover, (a) ^^=o (Ta)n cos nx is a Fourier series if and only if (7o)"->-0 at infinity 01, equivalently, the measure n is continuous at the origin, (b) 2™_i (Ta)" sin nx is a Fourier series if and only if the function x~'/i\iO. x)) is in ¿'[0,77]. These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients. G. H. Hardy first studied in [6] the transformation T. He showed that if 2«-o an cos nx is the Fourier series of a function in Lp[0, tt], then so is the series 2n-o (T~a)n cos nx for l^/><co, and the same is true for sine series.

The Hausdorff mean of a Fourier-Stieltjes transform

“…The objective here is to determine the Fourier analytic properties of T on the class of Fourier-Stieltjes transforms. Theorems 1 and 3 are the main results, which include the analogue of the results of Goes [7] and Georgakis [4], concerning the arithmetic means of Fourier-Stieltjes coefficients of a measure on the circle for the integral arithmetic average of the Fourier-Stieltjes transform of a measure on the real line. As a consequence, we obtain: (a) certain improved variants of the formulas of Fejer and Wiener for the inversion and quadratic variation of Fourier-Stieltjes transforms (Corollary 1); (b) a strengthened generalization of the mean ergodic theorem for a one-parameter group of unitary operators.…”

“…We may assume a > 0 on [0, oo). S<t(x) is an even function in LX{R) and nonincreasing for x > 0 ; its Hubert transform is odd and for x > 0, »/iW + AW-+A (4 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use…”

The Hausdorff Mean of a Fourier-Stieltjes Transform

“…Theorems 1 and 3 are the main results, which include the analogue of the results of Goes [7] and Georgakis [4], concerning the arithmetic means of Fourier-Stieltjes coefficients of a measure on the circle for the integral arithmetic average of the Fourier-Stieltjes transform of a measure on the real line. As a consequence, we obtain: (a) certain improved variants of the formulas of Fejer and Wiener for the inversion and quadratic variation of Fourier-Stieltjes transforms (Corollary 1); (b) a strengthened generalization of the mean ergodic theorem for a one-parameter group of unitary operators.…”