ON INTERPOLATION BY ALMOST TRIGONOMETRIC SPLINES

Sergey I Novikov

doi:10.15826/umj.2017.2.009

Abstract

The existence and uniqueness of an interpolating periodic spline defined on an equidistant mesh by the linear differential operator \({\cal L}_{2n+2}(D)=D^{2}(D^{2}+1^{2})(D^{2}+2^{2})\cdots (D^{2}+n^{2})\) with \(n\in\mathbb{N}\) are reproved under the final restriction on the step of the mesh. Under the same restriction, sharp estimates of the error of approximation by such interpolating periodic splines are obtained.

Highlights

Let D = d/dx, n ∈ N, and let L2n+2(D) = D2(D2 + 12)(D2 + 22) · · · (D2 + n2) (0.1)be the (2n + 2)th-order linear differential operator with constant real coefficients
The existence and uniqueness of periodic interpolating L-splines corresponding to an arbitrary linear differential operator with constant real coefficients were established in [10]
For N > n, we obtain a sharp estimate of the error of pointwise approximation by periodic interpolating almost trigonometric splines in the class of functions W∞(L2n+2) (Theorem 2)

Summary

Introduction

Be the (2n + 2)th-order linear differential operator with constant real coefficients. This term is not standard, and we use it only not to specify every time by what differential operator the considered splines are defined. For interpolation by polynomial splines, the existence, uniqueness and estimates of the error of approximation in many classes of functions are well-known (see, for instance, [1, Ch. V], [11], [12], [13], and references therein). The existence and uniqueness of periodic interpolating L-splines corresponding to an arbitrary linear differential operator with constant real coefficients were established in [10]. For N > n, we obtain a sharp estimate of the error of pointwise approximation by periodic interpolating almost trigonometric splines in the class of functions W∞(L2n+2) (Theorem 2). In the class W∞(L2n+2), the deviation from the periodic interpolating almost trigonometric splines is estimated by this function. The corresponding result was established by Nguen [5], [6, Ch. 2, §6]

Auxiliary results

Proofs of Theorems

Conclusion