A class of $L^{1}$-convergence
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- by R. Bojanić and Č. V. Stanojević PDF
- Trans. Amer. Math. Soc. 269 (1982), 677-683 Request permission
Abstract:
It is proved that if the Fourier coefficients $\{ {a_n}\}$ of $f \in {L^1}(0, \pi )$ satisfy $({\ast }){n^{ - 1}}\sum \nolimits _{k = n}^{2n} {{k^p}|\Delta {a_n}|p = o(1)}$, for some $1 < p \leqslant 2$, then $||{s_n} - f|| = o(1)$, if and only if ${a_n}\lg n = o(1)$. For cosine trigonometric series with coefficients of bounded variation and satisfying $({\ast })$ it is proved that a necessary and sufficient condition for the series to be a Fourier series is $\{ {a_n}\} \in \mathcal {C}$, where $\mathcal {C}$ is the Garrett-Stanojević [4] class.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 269 (1982), 677-683
- MSC: Primary 42A16; Secondary 42A20, 42A32
- DOI: https://doi.org/10.1090/S0002-9947-1982-0637717-3
- MathSciNet review: 637717