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On some complex spline operators

100%

Wronicz Z.

Opuscula Mathematica

2003

tom Vol. 23

99-115

The paper is concerned with the space Sn(ΔN) of splines in the complex (or real) variable z of degree n with respect to a given partition ΔN of a rectifiable Jordan curve Γ. We define an operator QN : LP(Γ) → Sn(ΔN), such that QN f = f for f ∈ Sn(ΔN), by means of a system of step functions "biorthogonal" to B-splines and then we estimate the order of approximation of f by QN f in the space Ck(Γ), k ≤ n. We apply the obtained results to approximation of analytic functions in the interior D of a Jordan curve Γ and of class Ck on D (k = 0,..., n - 1) by analytic splines defined in the interior Γ by means of the Cauchy integral. Then we consider the special case, where Γ is the interval [0, 1] and we estimate the order of approximation of f by QN f in the space Wnp([0, 1]) for 1 ≤ p ≤ ∞.

On some application of biorthogonal spline systems to integral equations

100%

Wronicz Z.

tom Vol. 25, no. 1

149-160

We consider an operator PN : LP(I) -> Sn(deltaN), such that PN f = f for f mem Sn(deltaN), where Sn(deltaN) is the space of splines of degree n with repect to a given partition deltaN of the interval I. This operator is defined by means of a system of step functions biorthogonal to B-splines. Then we use this operator to approximation to the solution of the Fredholm integral equation of the second kind. Convergence rates for the aproximation of the solution of this equation are given.

On a problem of Gevorkyan for the Franklin system

100%

Wronicz Z.

tom Vol. 36, no. 5

681--687

In 1870 G. Cantor proved that if [formula] for every real x, where [formula] then all coefficients cn are equal to zero. Later, in 1950 V.Ya. Kozlov proved that there exists a trigonometric series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. The conjecture remains open until now. In the present paper we show however that Kozlov's version remains true for Franklin's system.