József Szabados

Polynomial approximation on infinite intervals with weights having inner zeros

We generalize Laguerre weights on R+ by multiplying them by translations of finitely many Freud type weights which have singularities, and prove polynomial approximation theorems in the corresponding weighted spaces.

Approximation with exponential weights in (−1, 1)

Abstract We construct Shepard-type operators for weighted uniform approximation of functions with endpoint singularities by exponential-type weights in [−1,1]. Convergence theorems and pointwise and uniform approximation error estimates not possible by polynomials are proved.

Weighted approximation of functions on the real line by Bernstein polynomials

The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators.

The full Markov-Newman inequality for Müntz polynomials on positive intervals

For a function f defined on an interval [a, b] let ∥f∥ [a,b] := sup{|f(x)|: x ∈ [a,b]}. The principal result of this paper is the following Markov-type inequality for Muntz polynomials. Theorem. Let n > 1 be an integer. Let λ 0 , λ 1 ,...,λ n be n + 1 distinct real numbers. Let 0 < a < b. Then formula math. formula math. where the supremum is taken for all Q ∈ span{x λ 0,x λ 1,.., x λ n} (the span is the linear span over R).

A family of nonlinear difference equations

We study solutions ( x n ) n ? N of nonhomogeneous nonlinear second order difference equations of the type ? n = x n ( ? n , 1 x n + 1 + ? n , 0 x n + ? n , - 1 x n - 1 ) + ? n x n , n ? N , with?given?initial?data? { x 0 ? R & x 1 ? R + } where ( ? n ) n ? N ? R + & ( ? n , 0 ) n ? N ? R + & ( ? n ) n ? N ? R , and the left and right ? -coefficients satisfy either ( ? n , 1 ) n ? N ? R + & ( ? n , - 1 ) n ? N ? R + or ( ? n , 1 ) n ? N ? R 0 + & ( ? n , - 1 ) n ? N ? R 0 + . Depending on ones standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painleves discrete equation #1, that is, ? .

Generalized Bernstein polynomials with Pollaczek weight

Bernstein polynomials are a useful tool for approximating functions. In this paper, we extend the applicability of this operator to a certain class of locally continuous functions. To do so, we consider the Pollaczek weight w(x)@?exp(-1x(1-x)),0

The order of Lebesgue constant of Lagrange interpolation on several intervals

We consider Lagrange interpolation on the set of finitely many intervals. This problem is closely related to the least deviating polynomial from zero on such sets. We will obtain lower and upper estimates for the corresponding Lebesgue constant. The case of two intervals of equal lengths is simpler, and an explicit construction for two non-symmetric intervals will be given only in a special case.

On a quasi-interpolating Bernstein operator☆

We consider a special case of the modification of Lagrange interpolation due to Bernstein. Compared to Lagrange interpolation, these operators interpolate at less points, but they converge for all continuous functions in case of the Chebyshev nodes. Upper and lower estimates for the rate of convergence are given, and the saturation problem is partially solved.

Weighted Approximation of Functions with Inner Singularities by Exponential Weights in [-1,1]

Abstract We study the weighted uniform approximation of functions with inner singularities by exponential-type weights in [−1, 1]. Convergence theorems and pointwise and uniform approximation error estimates not possible by polynomials are proved.

Notes: Notes on a classic theorem of Erdős and Grünwald

We establish a companion result to a classic theorem of Erdos and Grunwald on the maximum of the fundamental functions of Lagrange interpolation based on the Chebyshev nodes.