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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ L\sp{1}$-convergence of Fourier series with complex quasimonotone coefficients


Author: Vera B. Stanojevic
Journal: Proc. Amer. Math. Soc. 86 (1982), 241-247
MSC: Primary 42A20; Secondary 42A32
MathSciNet review: 667282
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Abstract: A sequence of Fourier coefficients $ \left\{ {\hat f(n)} \right\}$ of a complex function in $ {L^1}(T)$ is said to be complex quasimonotone if there exists $ {\theta _0}$ such that

$\displaystyle \Delta \hat f(n) + \frac{\alpha } {n}\hat f(n) \in \left\{ {z\vert\left\vert {\arg z} \right\vert \leqslant {\theta _0} < \frac{\pi } {2}} \right\}$

for some $ \alpha \geqslant 0$ and for all $ n$. It is proved that Fourier series with asymptotically even and complex quasimonotone coefficients, satisfying

$\displaystyle \overline {\mathop {\lim }\limits_{n \to \infty } } \;{n^{1/q}}\m... ...ght\vert^{1/p}} = o(1),\; \lambda \to 1 + 0,\tfrac{1} {p} + \tfrac{1} {q} = 1, $

converges in $ {L^1}(T)$-norm if and only if $ \hat f(n)\lg \left\vert n \right\vert = o(1)$, $ n \to \infty $. A recent result of Č V. Stanojević [3] is a special case of the corollary of the main theorem.

References [Enhancements On Off] (What's this?)

  • [1] Časlav V. Stanojević, Tauberian conditions for $ {L^1}$-convergence of Fourier series, Trans. Amer. Math. Soc. 271 (1982), 237-244.
  • [2] William O. Bray and Časlav V. Stanojević, Tauberian $ {L^1}$-convergence classes of Fourier series. I, Trans. Amer. Math. Soc. (to appear).
  • [3] Časlav V. Stanojević, Classes of $ {L^1}$-convergence of Fourier and Fourier-Stieltjes series, Proc. Amer. Math. Soc. 82 (1981).
  • [4] M. Petrovic, Théorème sur les intégrales curvilignes, Math, de l'Univ. Beograd 2 (1933), 45-59.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0667282-1
PII: S 0002-9939(1982)0667282-1
Keywords: $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1982 American Mathematical Society