What question did this study set out to answer?

The aim is to generalize results on the absolute convergence of Fourier series to Fourier-Laplace series in specific weighted function spaces.

April 30, 2026Open Access

On the behavior of Fourier-Laplace series in L₏ (^m-1) classes

Key Points

The aim is to generalize results on the absolute convergence of Fourier series to Fourier-Laplace series in specific weighted function spaces.
Analyzed Fourier-Laplace series in the spaces $L_{p}(\sigma ^{m-1})$ for $1< p eq 2$.
Established conditions under which convergence occurs, extending previous analysis.
Provided sharpness of conditions specifically when $p = 2$.
Demonstrated that the convergence properties of Fourier-Laplace series hold under extended conditions for $1 < p eq 2$.
Showed that the results are sharp for $p = 2$, confirming the necessity of the established conditions.

Abstract

The recent contribution by Gogoladze and Meskhia (Proc. A. Razmadze Math. Inst. 141: 29–40, 2006) presents a generalization of classical results by Bernstein, Szász, Zygmund, and others on the absolute convergence of Fourier series with coefficients raised to a power ξ, analyzed via moduli of smoothness. In this work, we extend their findings to Fourier-Laplace series in the weighted function spaces L₏ (^m-1), 1< p 2, and we also establish the sharpness of our conditions when p = 2.

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