Tchakaloff’s theorem and k-integral polynomials in Banach spaces

Abstract

Tchakaloff’s theorem gives a quadrature formula for polynomials of a given degree with respect to a compactly supported positive measure which is absolutely continuous with respect to Lebesgue measure. We study the validity of two possible analogues of Tchakaloff’s theorem in an infinite-dimensional Banach space E: a weak form valid when E has a Schauder basis, and a stronger form requiring conditions on the support of the measure as well as on the space E.