On the $L_1$-convergence of Fourier series

S. Fridli

ArticleOriginal scientific text

Title

On the $L_{1}$ -convergence of Fourier series

Authors ¹

Affiliations

Deptartment of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, 1088 Budapest, Hungary

Abstract

Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier series in integral norm, almost everywhere, and if the function itself is in the real Hardy space, then also in the Hardy norm. We also compare it to the previously known conditions.

Keywords

ENG

Fourier series,

L_{1}

-convergence, a.e. convergence

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Title

On the L1-convergence of Fourier series

Affiliations

Abstract

Keywords

Bibliography

On the $L_{1}$ -convergence of Fourier series