P. Vértesi

Mean convergence of Hermite-Fejér interpolation

Abstract Weighted L p convergence of Hermite-Fejer interpolation based on the zeros of generalized Jacobi polynomials is investigated. The main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions.

A survey on mean convergence of interpolatory processes

Abstract A detailed account of what happened in the theory of mean convergence of Lagrange and Hermite-Fejer type interpolation in the last fifty-five years is given. Starting from the classical result of Erdős and Turan, (hopefully) all important developments are considered, in some cases with an indication of the method of proof. Even some yet unpublished results are included. A list of references helps to orientate those interested in the details.

Optimal lebesgue constant for lagrange interpolation

Fehlberg [Computing, 4 (1969), pp. 93–106] developed a family of eight-stage pairs of Runge–Kutta methods of orders 5 and 6. Subsequently, improved versions were derived independently by Butcher [“The Numerical Analysis of Ordinary Differential Equations,” John Wiley, New York, 1987] and Verner. Those obtained by Verner are only a slight generalization of Fehlberg’s methods, whereas those of Butcher have quite a different structure, and require one more stage when the high-order method is propagated. Prince and Dormand [J. Comput. Appl. Math., 7 (1981), pp. 67–76] have since constructed methods similar to, but more general than, those developed by Verner. This paper provides explicit formulas that relate the methods developed by Fehlberg, by Verner and by Dormand and Prince. In particular, each of Verner’s methods leads to a two-parameter family of Dormand–Prince methods. The formulas facilitate the study of these methods, which include the most effective known methods of orders 5 and 6. A particular set ...

On the Lebesgue function of weighted Lagrange interpolation. II

The aim of this paper is to continue our investigation of the Lebesgue function of weighted Lagrange interpolation by considering Erdős weights on ℝ and weights on [−1, 1]. The main results give lower bounds for the Lebesgue function on large subsets of the relevant domains.

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.

Hermite and Hermite-Fejér interpolations of higher order. II (mean convergence)

Weighted LP convergence of Hermite and Hermite-Fejér interpolations of higher order on the zeros of Jacobi polynomials is investigated. The results which cover the classical Hermite-Fejér case give necessary and in many cases sufficient conditions for such convergence for all continuous functions. Uniform convergence is considered, too.

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 the Lebesgue Function of Interpolation

Solving an old problem of P. Erdős, we prove the best possible in order estimation for the Lebesgue function of Lagrange interpolation.

On the L p convergence of Lagrange interpolating entire functions of exponential type

Abstract Let f: R ↦ C be a continuous, 2π-periodic function and for each n ϵ N let tn(f; ·) denote the trigonometric polynomial of degree ⩽n interpolating f in the points 2kπ (2n + 1) (k = 0, ±1, …, ±n) . It was shown by J. Marcinkiewicz that lim n → ∞ ∝ 0 2π ¦f(θ) − t n (f θ)¦ p dθ = 0 for every p > 0 . We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.

On the method of Somorjai

of Bal~zs where f~C[0, 1] and r > 2 is a fixed number. In [3] we tried to extend his argument for other interpolatory operators, but as F. Pint~r remarked, the proof ([3], Lemma 3.2]) is not correct. The aim of this paper is to prove a general saturation theorem which improves our previous attempt, too. We give our main resuIt in Section 2, an application in Section 3 and the proof in Sections 4 and 5. 2. Preliminaries and main theorem To make our theorem as general as possible we need the following ideas. A. Let I=[a , b] be a finite interval and let