On the convergence of periodic splines of arbitrary degree (Q582498)

The author considers an interpolation problem with 1-periodic polynomial splines of degree m with spline knots i/n (i\(\in {\mathbb{Z}}\); \(n>1)\). A new error estimate the following type is proved under the assumption that there exists a unique spline interpolation an arbitrary given 1-periodic function at the interpolation points (i-\(\lambda)\)/n (i\(\in {\mathbb{Z}}\); \(0\leq \lambda <1):\) If s is the 1-periodic spline interpolant of a given 1-periodic function \(f\in C^{m+1}({\mathbb{R}})\), then \(\| f^{(k)}- s^{(k)}\|_{\infty}=O(n^{-m-1+k})\) (0\(\leq k\leq m)\). This generalizes special results of \textit{F. Dubeau} and \textit{J. Savoie} [J. Approximation Theory 39, 77-88 (1983; Zbl 0516.41005) and IMA J. Numer. Anal. 5, 183-189 (1985; Zbl 0566.41029)]. Using the discrete Fourier transform and cyclic convolution, a numerical procedure is described.