The optimal reconstruction from blurred and nonuniformly sampled data based on the optimum discrete approximation minimizing various worst-case measures of error

Abstract

Extended interpolatory approximation is discussed for some classes of <i>n</i>-dimensional vector signals. Firstly, we present two sufficient conditions of the optimum approximation and prove that the proposed optimum approximation using fixed finite number of sample values satisfies these two conditions. Secondly, we discuss the optimum running approximation of <i>n</i>-dimensional time-limited vector signals based on a certain one-to-one correspondence between a vector signal and the corresponding vector error signal of approximation. The proposed optimum approximation has the minimum measure of error among almost all the linear and the nonlinear approximations using the same measure of error and generalized sample values. Note that the proposed optimum approximation can be realized by flexible FIR filter bank. The term "flexible" means that we can widely choose the number of paths and frequency response of time-invariant FIR analysis filters. Moreover, we can use sample points that are distributed on an arbitrary periodical pattern.