On a Tauberian theorem for the 𝐿¹-convergence of Fourier sine series

Bray, William O.

doi:10.1090/S0002-9939-1983-0691274-0

On a Tauberian theorem for the $L^{1}$-convergence of Fourier sine series
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by William O. Bray

Proc. Amer. Math. Soc. 88 (1983), 34-38

DOI: https://doi.org/10.1090/S0002-9939-1983-0691274-0

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Abstract:

In a recent Tauberian theorem of Stanojević [3] for the ${L^1}$-convergence of Fourier series, the notion of asymptotically even sequences is introduced. These conditions are satisfied if the Fourier coefficients $\{ \hat f(n)\}$ are even $(\hat f( - n) = \hat f(n))$, a case formally equivalent to cosine Fourier series. This paper applies the Tauberian method of Stanojević [3] separately to cosine and sine Fourier series and shows that the notion of asymptotic evenness can be circumvented in each case.

References

Sur l’ordre de grandeur des coefficients de la série de Fourier-Lebesque

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
Časlav V. Stanojević, Tauberian conditions for $L^{1}$-convergence of Fourier series, Trans. Amer. Math. Soc. 271 (1982), no. 1, 237–244. MR 648089, DOI 10.1090/S0002-9947-1982-0648089-2