Optimal interpolation formulas in the periodic function space of S.L. Sobolev

TL;DR

This paper develops explicit lattice optimal interpolation formulas in Sobolev spaces, calculates their error norms, and explores their connection to optimal quadrature formulas, supported by numerical results.

Contribution

It provides explicit formulas for lattice optimal interpolation in Sobolev spaces and links them to optimal quadrature formulas, advancing computational methods in this area.

Findings

Explicit formulas for interpolation coefficients are derived.

The error functional norms are calculated.

Numerical results demonstrate the effectiveness of the formulas.

Abstract

In this paper the problem of construction of lattice optimal interpolation formulas in the space $\widetilde{L_2^{(m)}} (0,1)$ is considered. Using S.L. Sobolev's method explicit formulas for the coefficients of lattice optimal interpolation formulas are given and the norm of the error functional of lattice optimal interpolation formulas is calculated. Moreover, connection between optimal interpolation formula in the space $\widetilde{L_2^{(m)}} (0,1)$ and optimal quadrature formula in this space is shown. Finally, numerical results are given.