On the arithmetic mean of Fourier-Stieltjes coefficients
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- by Constantine Georgakis PDF
- Proc. Amer. Math. Soc. 33 (1972), 477-484 Request permission
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.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- 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