There is No Free Lunch with Causal Approximations

This paper studies the approximation of continuous functions in subsets of all causal and stable transfer functions. Such approximations play a central roll in filter design, filter bank analysis, and in sampling, since any filtering can be considered as a kind of approximation in a space defined by the filters. The present paper studies in particular the consequences resulting from the causality and stability constrain imposed on the filter process. It is shown that there exists no linear approximation method which is also causal and stable. Only if either the causality or the stability constrain is left out, a linear approximation method may exist.