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ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the arithmetic mean of Fourier-Stieltjes coefficients


Author: Constantine Georgakis
Journal: Proc. Amer. Math. Soc. 33 (1972), 477-484
MSC: Primary 42A16
DOI: https://doi.org/10.1090/S0002-9939-1972-0298319-3
MathSciNet review: 0298319
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Abstract: Let $ \{ {a_n}\} _{n = 0}^\infty $ be the cosine Fourier-Stieltjes coefficients of the Borel measure $ \mu $ and $ \{ {a_0},({a_1} + \cdots + {a_n})/n\} _{n = 1}^\infty = \{ {(Ta)_n}\} _{n = 0}^\infty $ be the sequence of their arithmetic means. Then $ \sum\nolimits_{n = 0}^\infty {{{(Ta)}_n}\cos nx} $ is a Fourier-Stieltjes series. Moreover, (a) $ \sum\nolimits_{n = 0}^\infty {{{(Ta)}_n}\cos nx} $ is a Fourier series if and only if $ {(Ta)_n} \to 0$ at infinity or, equivalently, the measure $ \mu $ is continuous at the origin, (b) $ \sum\nolimits_{n = 1}^\infty {{{(Ta)}_n}\sin nx} $ is a Fourier series if and only if the function $ {x^{ - 1}}\mu ([0,x))$ is in $ {L^1}[0,\pi ]$. 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.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0298319-3
Keywords: Fourier-Stieltjes coefficients, Borel measure, discrete measure, inversion, arithmetic mean, Fourier coefficients, conjugate function, conjugate series
Article copyright: © Copyright 1972 American Mathematical Society

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