## Classes of $L^{1}$-convergence of Fourier and Fourier-Stieltjes series

HTML articles powered by AMS MathViewer

- by Časlav V. Stanojević PDF
- Proc. Amer. Math. Soc.
**82**(1981), 209-215 Request permission

## Abstract:

It is shown that the Fomin class ${\mathcal {F}_p}(1 < p \leqslant 2)$ is a subclass of $\mathcal {C} \cap \mathcal {B}\mathcal {V}$, where $\mathcal {C}$ is the Garrett-Stanojević class and $\mathcal {B}\mathcal {V}$ the class of sequences of bounded variation. Wider classes of Fourier and Fourier-Stieltjes series are found for which ${a_n}\;{\text {lg}}\;n = o(1),n \to \infty$, is a necessary and sufficient condition for ${L^1}$-convergence. For cosine series with coefficients in $\mathcal {B}\mathcal {V}$ and $n\Delta {a_n} = O(1)$, $n \to \infty$, necessary and sufficient integrability conditions are obtained.## References

- S. A. Teljakovskiĭ,
*A certain sufficient condition of Sidon for the integrability of trigonometric series*, Mat. Zametki**14**(1973), 317–328 (Russian). MR**328456**
A. N. Kolmogorov, - John W. Garrett and Časlav V. Stanojević,
*Necessary and sufficient conditions for $L^{1}$ convergence of trigonometric series*, Proc. Amer. Math. Soc.**60**(1976), 68–71 (1977). MR**425480**, DOI 10.1090/S0002-9939-1976-0425480-3 - 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 - G. A. Fomin,
*A class of trigonometric series*, Mat. Zametki**23**(1978), no. 2, 213–222 (Russian). MR**487218** - S. A. Teljakovskiĭ and G. A. Fomin,
*Convergence in the $L$ metric of Fourier series with quasimonotone coefficients*, Trudy Mat. Inst. Steklov.**134**(1975), 310–313 (Russian). Theory of functions and its applications (Collection of articles dedicated to Sergeĭ Mihaĭlovič Nikol′skiĭ on the occasion of his seventieth birthday). MR**0394013** - 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

*Sur l’ordre de grandeur des coefficients de la série de Fourier-Lebesgue*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (1923), 83-86. S. Sidon,

*Hinreichende Bedingungen fur den Fourier-charakter einer trigonometrischen Reihe*, J. London Math. Soc.

**14**(1939), 158-160.

## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**82**(1981), 209-215 - MSC: Primary 42A32; Secondary 42A20
- DOI: https://doi.org/10.1090/S0002-9939-1981-0609653-4
- MathSciNet review: 609653