Interpolation Splines Minimizing a Semi-Norm in a Sobolev Hilbert Space

Abstract

Background: The interpolation problem is one of the central problems of approximation theory. Its classical solution relies on the interpolation polynomial; however, polynomial approximation is inefficient for functions of finite and low smoothness, which occur most often in applications. Splines are therefore used, understood in the variational sense as elements of a Hilbert space that minimize a certain functional. For spaces generated by linear differential operators with constant coefficients such splines are constructed by Sobolev’s method, which rests upon discrete analogues of differential operators.

Purpose: To present the general construction of the interpolation spline minimizing the semi-norm of the space in question for an arbitrary odd order of the differential operator, and to show that the problem corresponding to the first odd order is a particular case of this construction admitting a fully explicit solution.

Method: The variational problem is reduced to a system of linear equations for the jumps of the highest derivative of the spline; the system is rewritten in convolution form in terms of discrete argument functions, and its solution is recovered by convolution with the discrete analogue of the corresponding differential operator of even order.

Results: For the first odd order the discrete analogue is obtained in a closed threepoint form containing no decaying geometric part, which makes it possible to write the coefficients of the spline in explicit elementary form. The spline is exact on the exponential functions spanning the kernel of the operator; its coefficients at the interior nodes agree with previously known results, whereas the spline itself differs from the known one. Numerical experiments confirm this exactness and the second order of convergence on smooth functions of general form.

Implication: The formulas complete the family of exponential interpolation splines at the first odd level and yield an algorithm that is simple to implement.