The divergence behavior of adaptive signal processing algorithms with finite search horizon

Many important non-adaptive approximation methods are know to diverge for almost all functions from certain Banach space X. One can show that a corresponding adaptive method will improve this behavior in the sense that it converges to the desired result for almost all functions in X. However, even though an adaptive method tries to find an optimal approximation for any given function, the search horizon (i.e. the search set) has to be finite in practical applications. This paper shows that an adaptive method with finite search horizon either converges for all f ϵ X or it diverges for almost all f ϵ X. As an example, we show that there exists no realizable adaptive method which can calculate the Hilbert transform of a continuous function f based on samples of f.