Gauss interpolation formulas and totally positive kernels

This paper simplifies and generalizes an earlier result of the author’s on Gauss interpolation formulas for the one-dimensional heat equation. Such formulas approximate a function at a point $({x^\ast },{t^\ast })$ in terms of a linear combination of its values on an initial-boundary curve in the (x, t) plane. The formulas are characterized by the requirement that they be exact for as many basis functions as possible. The basis functions are generated from a Tchebycheff system on the line $t = 0$ by an integral kernel $K(x,y,t)$, in analogy with the way heat polynomials are generated from the monomials ${x^i}$ by the fundamental solution to the heat equation. The total positivity properties of $K(x,y,t)$ together with the theory of topological degree are used to establish the existence of the formulas.