Integral representation of slowly growing equidistant splines |
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Authors: | Valery A Zheludev |
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Institution: | (1) School of Mathematical Sciences, Tel Aviv University, 69978 Ramat Aviv, Tel Aviv, Israel |
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Abstract: | In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|s). We present an integral representation of such splines with a distribution kernel. This representation is related to the
Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential
splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines.
First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form
a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations
form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline
spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting
a function onto the spline space and establish therewith exact estimations of the approximation errors. |
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