On the behavior of Fourier-Laplace series in Lₚ(σ ᵐ⁻¹) classes

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 Lₚ(σ ᵐ⁻¹), 1< p≤ 2, and we also establish the sharpness of our conditions when $p = 2$.