Existence of Gauss Interpolation Formulas for the One-Dimensional Heat Equation

Abstract

Abstract. Let C = {(x(s), t(s)): a < i < b} be a Jordan arc in the x-t plane satisfying (x(a), f(a)) = (a, f.), (x(b), t(b)) = (b, f "), and t(s) < tt when a < s < b. Let a < xt < b. We prove the existence of Gauss interpolation formulas for C and the point (x" f"), for solutions u of the one-dimensional heat equation, ut = uxx. Such formulas approximate u(xt, tt) in terms of a linear combination of its values on C.The formulas are characterized by the requirement that they are exact for as many basis functions (the heat polynomials) as possible.