Construction of interpolation splines minimizing the semi-norm in the space $K_2(P_m)$

TL;DR

This paper constructs interpolation splines that minimize a specific semi-norm in the $K_2(P_m)$ space using Sobolev's method, providing explicit formulas and ensuring exactness for certain polynomials and trigonometric functions.

Contribution

It introduces a new class of interpolation splines minimizing a semi-norm in $K_2(P_m)$ with explicit formulas and proven exactness for specific functions.

Findings

Explicit formulas for spline coefficients are derived.

The spline is exact for polynomials up to degree $m-3$.

The spline exactly interpolates sine and cosine functions.

Abstract

In the present paper, using S.L. Sobolev's method, interpolation spline that minimizes the expression $\int_0^1(\varphi^{(m)}(x)+\omega^2\varphi^{(m-2)}(x))^2dx$ in the $K_2(P_m)$ space are constructed. Explicit formulas for the coefficients of the interpolation splines are obtained. The obtained interpolation spline is exact for monomials $1,x,x^2,..., x^{m-3}$ and for trigonometric functions $\sin\omega x$ and $\cos\omega x$.