in the case of half integerα and it is expressed in terms of the matrix coefficients determining the linear summation method. The author also proves the
analogue of the well-known theorem by S. M. Nikol'skii on the necessary and sufficient condition for the summability of trigonometric
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.
is also valid, where (dk) is a logarithmic weight sequence and DN = ∑k=1Ndk. In this paper we extend the last result for a very large class of weight sequences (dk), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory,
are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with,
The aim of this paper is to continue our investigations started in , where we studied the summability of weighted Lagrange
interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly
convergent in a suitable Banach space (Cρ, ‖‖ρ) of continuous functions (ρ denotes (another) weight). In  we formulated several conditions with respect to w, ρ, (Cρ, ‖‖ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when
w and ρ are Freud-type weights. We shall show that the conditions of results of  hold in this case. The order of convergence
will also be considered.