On the conditioning of polynomial histopolation

Histopolation is the approximation procedure that associates a degree $d-1$ polynomial $p_{d-1}\in {\mathcal{P}}_{d-1}(I)$ 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.