Error bounds for interpolation with piecewise exponential splines of order two and four

TL;DR

This paper derives explicit pointwise error bounds for interpolating smooth functions using piecewise exponential splines of orders two and four, extending known estimates for cubic splines to a broader class relevant for multivariate polysplines.

Contribution

It provides new explicit error bounds for piecewise exponential spline interpolation of order four, extending classical cubic spline estimates to this more general class.

Findings

Error bounds for order four exponential spline interpolation are established.

Estimates for order two exponential splines are used inductively to derive bounds for order four.

The results are relevant for multivariate polyspline construction.

Abstract

Explicit pointwise error bounds for the interpolation of a smooth function by piecewise exponential splines of order four are given. Estimates known for cubic splines are extended to a natural class of piecewise exponential splines which are appearing in the construction of multivariate polysplines. The error estimates are derived in an inductive way using error estimates for the interpolation of a smooth function by exponential splines of order two.