Approximation by Splines of Hermite Type

Abstract

The approximation evaluations by polynomial splines are well-known. They are obtained by the similarity principle; in the case of non-polynomial splines the implementation of this principle is difficult. Another method for obtaining of the evaluations was discussed earlier (see [1]) in the case of nonpolynomial splines of Lagrange type. The aim of this paper is to obtain the evaluations of approximation by non-polynomial splines of Hermite type. Considering a linearly independent system of column-vectors , . Let be square matrix. Supposing that and are columns with components from the linear space such that . Let be vector with components belonging to conjugate space . For an element we consider a linear combination of elements By definition, put . The discussions are based on the next assertion. The following relation holds : where the second factor on the right-hand side is the determinant of a block-matrix of order m + 2. Using this assertion, we get the representation of residual of approximation by minimal splines of Hermite type. Taking into account the representation, we get evaluations of the residual and calculate relevant constants. As a result the obtained evaluations are exact ones for components of generated vector-function .