Acuity of Observation of the Heat Equation on a Bounded Domain

Abstract

In [1], Gilliam, Li and Martin showed that the heat equation on a bounded domain in ℝⁿ is discretely observable under certain general conditions. Their sampling method was to sample atp points in the region, wherep is the largest multiplicity of any eigenvalue in the corresponding eigenvalue problem, and to sample an infinite number of times. Of course, in practice one samples a finite number of times and then reconstructs an approximate solution to the equation. In this paper we investigate the accuracy of this process and show how the error in the estimate depends on the tail of the Fourier expansion of the initial condition. In some cases we can show that the size of the tail depends, in turn, on the smoothness of the initial condition.