On a theorem of P. L. Uljanov
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- by Vera B. Stanojevic PDF
- Proc. Amer. Math. Soc. 90 (1984), 370-372 Request permission
Abstract:
It is shown that if $c\left ( n \right ) = o\left ( 1 \right )$, $\left | n \right | \to \infty$, and ${\sum _{\left | n \right |}}_{ < \infty }\left | {{\Delta ^m}c\left ( n \right )} \right | < \infty$, for some integer $m \geqslant 1$, then the series ${\sum _{\left | n \right |}}_{ < \infty }c\left ( n \right ){e^{\operatorname {int} }}$ converges to some $f \in {L^p}\left ( {\mathbf {T}} \right )$ for any $0 < p < 1 / m$.References
- P. L. Ul′yanov, Application of $A$-integration to a class of trigonometric series, Mat. Sb. N.S. 35(77) (1954), 469–490 (Russian). MR 0065680
- J. W. Garrett, C. S. Rees, and Č. V. Stanojević, $L^{1}$-convergence of Fourier series with coefficients of bounded variation, Proc. Amer. Math. Soc. 80 (1980), no. 3, 423–430. MR 580997, DOI 10.1090/S0002-9939-1980-0580997-7
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 90 (1984), 370-372
- MSC: Primary 42A20; Secondary 42A32
- DOI: https://doi.org/10.1090/S0002-9939-1984-0728350-0
- MathSciNet review: 728350