F Frans Schurer

On the degree of approximation by the operators of de la Vallée Poussin

In this paper we study the degree of approximation of functionsf inC2π andC2π1 by the operatorsVn ofde la Vallée Poussin. The quality of approximation is measured in terms of the modulus of continuity off andf′ respectively. Forn∈ℕ so-called exact constants of approximation are determined. Furthermore, the asymptotic behaviour of these constants is investigated asn→∞.

On the degree of approximation of functions in C2π1 with operators of the Jackson type

Abstract Let C 2 π 1 be the class of real functions of a real variable that are 2π-periodic and have a continuous derivative. The positive linear operators of the Jackson type are denoted by L n , p ( n ∈ N ), where p is a fixed positive integer. The object of this paper is to determine the exact degree of approximation when approximating functions f ϵ C 2 π 1 with the operators L n , p . The value of max x ¦L n,p (f x) − f(x)¦ is estimated in terms of ω 1 ( f ; δ ), the modulus of continuity of f ′, with δ = π n . Exact constants of approximation are obtained for the operators L n , p (n ∈ N , p ≥ 2) and for the Fejer operators L n ,1 (n ∈ N ). Furthermore, the limiting behaviour of these constants is investigated as n → ∞, and p → ∞, separately or simultaneously.

Euler L-splines and an extremal problem for periodic functions

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