Approximation by Splines of Hermite Type

Affiliation(s)

Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia.

Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia.

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 .

Cite this paper

Y. Dem’yanovich and I. Burova, "Approximation by Splines of Hermite Type,"*Applied Mathematics*, Vol. 4 No. 11, 2013, pp. 5-10. doi: 10.4236/am.2013.411A3002.

Y. Dem’yanovich and I. Burova, "Approximation by Splines of Hermite Type,"

References

[1] Yu. K. Dem’yanovich, “Approximation by Minimal Splines,” Journal of Mathematical Sciences, Vol. 193, No. 2, 2013, pp. 261-266.

[2] I. G. Burova and Yu. K. Dem’yanovich, “Theory of Minimal Splines,” St.-Petersburg University Press, St.-Petersburg, 2000.

[3] A. O. Gelfond, “Calculation of Finite Differences,” Nauka Press, Moscow, 1967.