Acuity of Observation for Invariant Evolution Equations

Abstract

In a recent paper [4] we studied discrete observability for invariant evoLution equations on compact homogeneous spaces, e.g. for the heat equation on the sphere. The observations there were given by simultaneous measurements, corresponding to function evaLuations. The initial data was observed as a limit of truncations deduced from a finite number of measurements. That procedure naturally involves two types of errors. First, observations qua evaLuation functionals are restricted to a finite part of the Fourier Peter Weyl expansion; that restriction implicitly involves a convoLution. See (1.4) and (1.6) below. We think of the resulting error as the error in the head of the approximation. Second, the actual initial data minus the truncations are the usual type of error terms; we think of them as the error in the tail of the approximation. In this paper we show that the error in the head depends linearly on the error in the tail. We then investigate the extent to which smoothness of the initial data function controls the tail error through a set of Sobolev inequalities. We also investigate consequences of polynomial spectral growth conditions on the rate of vanishing of the tail error. Finally, we specialize these results to riemannian symmetric spaces.