Upper bounds for the error in some interpolation and extrapolation designs

TL;DR

This paper establishes probabilistic upper bounds for the error in estimating analytic functions using interpolation and extrapolation designs, focusing on how design choices affect estimation accuracy and required observations.

Contribution

It provides a rule for the minimal number of observations needed to ensure a specified error bound with high probability in interpolation and extrapolation of analytic functions.

Findings

Derived probabilistic upper bounds for estimation error.

Identified how design parameters influence the number of observations needed.

Provided guidelines for optimal observation strategies.

Abstract

This paper deals with probabilistic upper bounds for the error in functional estimation defined on some interpolation and extrapolation designs, when the function to estimate is supposed to be analytic. The error pertaining to the estimate may depend on various factors: the frequency of observations on the knots, the position and number of the knots, and also on the error committed when approximating the function through its Taylor expansion. When the number of observations is fixed, then all these parameters are determined by the choice of the design and by the choice estimator of the unknown function. The scope of the paper is therefore to determine a rule for the minimal number of observation required to achieve an upper bound of the error on the estimate with a given maximal probability.