On weighted integrability of trigonometric series and 𝐿¹-convergence of Fourier series

Bray, William O.; Stanojević, Časlav V.

doi:10.1090/S0002-9939-1986-0813809-X

On weighted integrability of trigonometric series and $L^ 1$-convergence of Fourier series
HTML articles powered by AMS MathViewer

by William O. Bray and Časlav V. Stanojević

Proc. Amer. Math. Soc. 96 (1986), 53-61

DOI: https://doi.org/10.1090/S0002-9939-1986-0813809-X

PDF | Request permission

Abstract:

A result concerning integrability of $f(x)L(1/x)(g(x)L(1/x))$, where $f(x)(g(x))$ is the pointwise limit of certain cosine (sine) series and $L( \cdot )$ is slowly vary in the sense of Karamata [5] is proved. Our result is an excluded case in more classical results (see [4]) and also generalizes a result of G. A. Fomin [1]. Also a result of Fomin and Telyakovskii [6] concerning ${L^1}$-convergence of Fourier series is generalized. Both theorems make use of a generalized notion of quasi-monotone sequences.

References

G. A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), no. 2, 213–222 (Russian). MR 487218
William O. Bray, On a theorem of Stanojević, Proc. Amer. Math. Soc. 83 (1981), no. 1, 59–62. MR 619981, DOI 10.1090/S0002-9939-1981-0619981-4
William O. Bray and Vera B. Stanojevic, On the integrability of complex trigonometric series, Proc. Amer. Math. Soc. 93 (1985), no. 1, 51–58. MR 766526, DOI 10.1090/S0002-9939-1985-0766526-8
Ralph P. Boas Jr., Integrability theorems for trigonometric transforms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 38, Springer-Verlag New York, Inc., New York, 1967. MR 0219973, DOI 10.1007/978-3-642-87108-5

Sur un mode de croissance reguliere des fonctions

4

On convergence in

-metric of Fourier series with quasi-monotone coefficients

134

J. W. Garrett, C. S. Rees, and Č. V. Stanojević, On $L^{1}$ convergence of Fourier series with quasi-monotone coefficients, Proc. Amer. Math. Soc. 72 (1978), no. 3, 535–538. MR 509250, DOI 10.1090/S0002-9939-1978-0509250-5
Otto Szász, Quasi-monotone series, Amer. J. Math. 70 (1948), 203–206. MR 22917, DOI 10.2307/2371946

Sur certains "Tauberian Theorems" de M. M. Hardy et Littlewood

3

Slowly varying functions with remainder term and their applications in analysis

Friedrich Riesz, Über eine Verallgemeinerung der Parsevalschen Formel, Math. Z. 18 (1923), no. 1, 117–124 (German). MR 1544624, DOI 10.1007/BF01192400