On the Estimation of Strong Means of Fourier Series

Abstract

We study problem of $(λ, φ)$ -strong summation of number series by the regular method $λ$ with power summation of the function $φ$. The accumulated results are extended to the case of Fourier expansions in trigonometric functions $f ϵ L_p, p > 1$, where $C$ is the set of $2π$-periodic continuous functions. Some results are also obtained for the estimation of strong means of the method $λ$ in $L_p, p > 1$, at the Lebesgue point $x$ of the function $f$ under certain additional conditions in the case where the function $φ$ tends to infinity as $u → ∞$ faster than the exponential function $\exp (βu) − 1, β > 0$.