On the Approximation of $L_{2}$ Inner Products From Sampled Data

Most signal processing applications are based on discrete-time signals although the origin of many sources of information is analog. In this paper, we consider the task of signal representation by a set of functions. Focusing on the representation coefficients of the original continuous-time signal, the question considered herein is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a bounded operator, a vector-like interpretation for this approximation problem has been derived, giving rise to an optimal discrete approximation scheme different from the Riemann-type sum often used. The objective of this optimal scheme is in the min-max sense and no bandlimitedness constraints are imposed. Tight upper bounds on this optimal and the Riemann-type sum approximation schemes are then derived. We further consider the case of a finite number of samples and formulate a closed-form solution for such a case. The results of this work provide a tool for finding the optimal scheme for approximating an L₂ inner product, and to determine the maximum potential representation error induced by the sampling process. The maximum representation error can also be determined for the Riemann-type sum approximation scheme. Examples of practical applications are given and discussed