Spline approximation methods with uniform meshes in algebras of multiplication and convolution operators (Q2774475)

The theme of this paper is stability of spline approximation methods on uniform lattices. A characterisation theorem for stability of maximum defect spline approximation is obtained by proving equivalence of invertibility of a certain sequence in a \(C^*\)-algebra and the applicability of the spline approximation method to a given operator. That operator is a multiplication or convolution operator with multiplication or convolution by piecewise continuous functions, respectively. The applications which are given to demonstrate the usefulness of the result are Galerkin methods and collocation methods that are used to solve differential equations.