L. Szili

On Uniform Convergence of Sequences of Certain Linear Operators

The paper is devoted to the study of the uniform convergence of φ-summations of Fourier series and discrete Fourier series. We show that by choosing different parameters of these operators different orders of the uniform convergence can be attained on the space C2π.

On multivariate projection operators

This paper deals with multivariate Fourier series considering triangular type partial sums. Among others we give the exact order of the corresponding operator norm. Moreover, a generalization of the so-called Faber-Marcinkiewicz-Berman theorem has been proved.

On the Summability of Trigonometric Interpolation Processes

The aim of this paper is to show that several processes studied in trigonometric interpolation theory can be obtained by φ-sums of discrete Fourier series. We shall investigate the uniform convergence of the sequences of thesepolynomials. We show that the convergence of several processes can be seen immediately from suitable explicit forms of the corresponding polynomials. Error estimates for the approximation can be also obtained by certain general results.

An Erdős type convergence process in weighted interpolation. II

The aim of this paper is to continue the investigation of the second author started in [14], where a weighted version of a classical result of P. Erdős was proved using Freud type weights. We shall show that an analogous statement is true for weighted interpolation if we consider exponential weights on [-1,1].

On summability of weighted Lagrange interpolation. II

The aim of this paper is to continue our investigations started in [15], 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 [15] 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 [15] hold in this case. The order of convergence will also be considered.

On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials

The aim of this paper is to investigate the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials. Starting from the Lagrange interpolation polynomials, we shall construct a wide class of discrete processes which are uniformly convergent in a suitable weighted space of continuous functions. Error estimates for the approximation will also be considered.

Some Erdös-type Convergence Processes in Weighted Interpolation

In 1943, P. Erdos [5] showed that if the interpolation point system X n C [-1, 1] (n ∈ N) such that the fundamental polynomials of Lagrange interpolation are uniformly bounded in [-1, 1] then for every f∈ C[-1, 1] and c > 0 there exists a sequence of polynomials ϕ n of degree ≤ n(l+c) (n ∈ N) which interpolates f at the points X n and it tends to f uniformly in [-1, 1]. The weighted versions of this result were proved in [19] and [18] using Freud-type weights and exponential weights on [-1, 1]. The aim of this paper is to show that analogue statements are true for weighted interpolation if we consider Erdos-type and some ultraspherical weights.

Full length article: Lp-convergence of Hermite and Hermite-Fejér interpolation

Necessary and sufficient conditions for the weighted L^p-convergence of Hermite and Hermite-Fejer interpolation of higher order based on Jacobi zeros are given, extending previous results for Lagrange interpolation. Error estimates in the weighted L^p-norm are also shown.