Error Estimates for the Cardinal Spline Interpolation

Abstract

For the Sobolev class $W_{\textrm{per}}^{m,\infty}(\mathbb{R})$ of 1-periodic functions, an unimprovable error estimate for the spline interpolants of order $m$ on the uniform grid is known. In the present paper, this error estimate is extended to the Sobolev class $V^{m,\infty}(\mathbb{R})$ of (nonperiodic) functions on $\mathbb{R}$ having bounded $m$th derivative. Some further error estimates are established including the error estimates for derivatives of the spline interpolant.