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Transactions of the American Mathematical Society

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Tauberian conditions for $ L\sp{1}$-convergence of Fourier series


Author: Časlav V. Stanojević
Journal: Trans. Amer. Math. Soc. 271 (1982), 237-244
MSC: Primary 42A16; Secondary 42A20
DOI: https://doi.org/10.1090/S0002-9947-1982-0648089-2
MathSciNet review: 648089
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Abstract: It is proved that Fourier series with asymptotically even coefficients and satisfying $ {\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}\vert\Delta \hat f(j){\vert^p} = 0$, for some $ 1 < p \leqslant 2$, converge in $ {L^1}$-norm if and only if $ \vert\vert\hat f(n){E_n} + \hat f( - n){E_{ - n}}\vert\vert = o(1)$, where $ {E_n}(t) = \sum _{k = 0}^n{e^{ikt}}$. Recent results of Stanojević [1], Bojanic and Stanojević [2], and Goldberg and Stanojević [3] are special cases of some corollaries to the main theorem.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0648089-2
Keywords: $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1982 American Mathematical Society