The optimum approximation in generalized time-frequency domains and application to numerical simulation of partial differential equations

Extended optimum interpolatory approximation is presented for a certain set of signals having n variables. As the generalized spectrum of a signal, we consider a v-dimensional vector. These variables can be contained in one of the time domain, the frequency domain or the time-frequency domain. Sometimes, these can be contained in the space-variable domain or in the space-frequency variable domain. To construct the theory across these domains, we assume that the number of variables for a signal and its generalized spectrum are different, in general. Under natural assumption that those generalized spectrums have weighted norms smaller than a given positive number, we prove that the presented approximation has the minimum measure of approximation error among all the linear and the nonlinear approximations using the same generalized sample values. Application to numerical simulation of partial differential equations is considered. In this application, a property for discrete orthogonality associated with the presented approximation plays an essential part.