Necessary and sufficient conditions for 𝐿¹ convergence of trigonometric series

Garrett, John W.; Stanojević, Časlav V.

doi:10.1090/S0002-9939-1976-0425480-3

Necessary and sufficient conditions for $L\sp{1}$ convergence of trigonometric series

Authors: John W. Garrett and Časlav V. Stanojević
Journal: Proc. Amer. Math. Soc. 60 (1976), 68-71
MSC: Primary 42A20
DOI: https://doi.org/10.1090/S0002-9939-1976-0425480-3
MathSciNet review: 0425480
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Abstract: It is shown that for the class of cosine series satisfying $a(n)\log n = o(1)$ and $\Delta a(n) \geqslant 0$ that integrability and ${L^1}$ convergence occur together. Relaxing the monotonicity to bounded variation we show that our previous result cannot be extended.

[1] John W. Garrett and Časlav V. Stanojević, On 𝐿¹ convergence of certain cosine sums, Bull. Amer. Math. Soc. 82 (1976), no. 1, 129–130. MR 394001, https://doi.org/10.1090/S0002-9904-1976-13990-0
[2] S. A. Teljakovskiĭ, A certain sufficient condition of Sidon for the integrability of trigonometric series, Mat. Zametki 14 (1973), 317–328 (Russian). MR 328456