On the conditioning of polynomial histopolation
This addresses numerical stability issues in approximation theory for researchers in computational mathematics, but it is incremental as it builds on known Vandermonde matrix results.
The paper tackled the conditioning of polynomial histopolation matrices as their size increases, showing that using the monomial basis leads to exponential conditioning, while Chebyshev polynomials of the second kind can achieve bounded conditioning with appropriate supports.
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.