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On some application of biorthogonal spline systems to integral equations

Wronicz Z.

Opuscula Mathematica

2005

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 some complex spline operators

Wronicz Z.

Opuscula Mathematica

2003

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 ≤ ∞.