Polynomial Interpolation of Function Averages on Interval Segments

TL;DR

This paper investigates polynomial interpolation based on function averages over segments, exploring its mathematical properties, solution methods, and numerical stability, with applications to differential forms.

Contribution

It introduces a novel interpolation approach using segment averages, providing theoretical analysis, explicit basis systems, and numerical conditioning insights.

Findings

Established conditions for unisolvence

Derived explicit Lagrange-type basis systems

Provided bounds on the Lebesgue constant for stability

Abstract

Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyse fundamental mathematical properties of this problem as existence, uniqueness and numerical conditioning of its solution. We will provide concrete conditions for unisolvence, explicit Lagrange-type basis systems for its representation, and a numerical method for its solution. To study the numerical conditioning, we will provide concrete bounds of the Lebesgue constant in a few distinguished cases.