On the conditioning of polynomial histopolation

TL;DR

This paper investigates the mathematical properties of histopolation, focusing on the unisolvence and conditioning of associated matrices, revealing exponential conditioning with monomials and bounded conditioning with Chebyshev polynomials.

Contribution

It provides an asymptotic analysis of matrix conditioning in histopolation, highlighting the impact of basis choice on numerical stability.

Findings

Conditioning is exponential with monomial basis as size increases.

Chebyshev polynomial basis yields bounded conditioning.

Frobenius norm exhibits linear growth with matrix size.

Abstract

Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$ coincides with that of $f$ on a collection of $ d $ distinct segments $s_i$. In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size $d$ tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of $d$ when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev polynomials of second kind exhibit a bounded conditioning. A linear behavior is also observed in the Frobenius norm.