On necessary and sufficient conditions of the regularity of summation methods for Laguerre-Fourier series

摘要

The problem of convergence of linear means is considered for the Laguerre-Fourier series of continuous functions. An upper estimate is obtained for the Laguerre-Lebesgue function in terms of the entries of the matrix which determines the linear summability method in question. This allows us to prove for such series an analogue of the well-known theorem by S. M. Nikol'skii which provides necessary and sufficient conditions for the summability of trigonometric Fourier series. A theorem on the regularity of the summability methods is also established.