Analytic and numerical aspects of the observation of the heat equation

We present a simple and extremely accurate procedure for approximating initial temperature for the heat equation on the line using a discrete time and spatial sampling. The procedure is based on the "sinc expansion" which for functions in a particular class yields a uniform exponential error bound with exponent depending on the number of spatial sample locations chosen. Further the temperature need only be sampled at one and the same temporal value for each of the spatial sampling points. For N spatial sample points, the approximation is reduced to solving a linear system with a (2N + 1) ?? (2N + 1) coefficient matrix. This matrix is a symmetric toeplitz matrix and hence is determined by computing only 2N + 1 values using quadrature.