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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Tauberian conditions for $L^{1}$-convergence of Fourier series
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by Časlav V. Stanojević PDF
Trans. Amer. Math. Soc. 271 (1982), 237-244 Request permission


It is proved that Fourier series with asymptotically even coefficients and satisfying ${\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}|\Delta \hat f(j){|^p} = 0$, for some $1 < p \leqslant 2$, converge in ${L^1}$-norm if and only if $||\hat f(n){E_n} + \hat f( - n){E_{ - n}}|| = o(1)$, 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.
  • Časlav V. Stanojević, Classes of $L^{1}$-convergence of Fourier and Fourier-Stieltjes series, Proc. Amer. Math. Soc. 82 (1981), no. 2, 209–215. MR 609653, DOI 10.1090/S0002-9939-1981-0609653-4
  • R. Bojanić, An estimate for the rate of convergence of a general class of orthogonal polynomial expansions of functions of bounded variation, Mathematical analysis and its applications (Kuwait, 1985) KFAS Proc. Ser., vol. 3, Pergamon, Oxford, 1988, pp. 1–16. MR 951652
  • R. R. Goldberg and Č. V. Stanojević, ${L^1}$-convergence and Segal algebras of Fourier series, preprint (1980). J. Karamata, Teorija i praksa Stieltjes-ova integrala, Srpska Akademija Nauka, Beograd, 1949. —, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53. G. H. Hardy, Theorems relating to the convergence and summability of slowly oscillating series, Proc. London Math. Soc. Ser. 2 8 (1909).
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 271 (1982), 237-244
  • MSC: Primary 42A16; Secondary 42A20
  • DOI:
  • MathSciNet review: 648089