Sharp asymptotics of the Lp approximation error for interpolation on block partitions

TL;DR

This paper derives precise asymptotic estimates for the error of spline interpolation on block partitions in multiple dimensions, analyzing the constants involved and considering various projection operators.

Contribution

It provides sharp asymptotic error estimates for spline interpolation on block partitions, including explicit constants and analysis of different projection operators.

Findings

Sharp asymptotic error estimates for spline interpolation

Explicit formulas for asymptotic constants in certain cases

Analysis of various projection operators' effects on error

Abstract

Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in IRd. We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases.