Higher Order Approximations in the Heat Equation and the Truncated Moment Problem

In this paper we employ linear combinations of n heat kernels to approximate solutions to the heat equation. We show that such approximations are of order O(t ( 1 2p− 2n+1 2 ) in Lp-norm, 1 ≤ p ≤ ∞, as t → ∞. For positive solutions of the heat equation such approximations are achieved using the theory of truncated moment problems. For general sign-changing solutions, this type of approximations are obtained by simply adding an auxiliary heat kernel. Furthermore, inspired by numerical computations, we conjecture that such approximations converge geometrically as n→∞ for any fixed t > 0.