Orthogonal expansions and the causality law

Abstract

The use of orthogonal expansions leads sometimes to results that clearly do not satisfy the causality law. The best known example is the filtering of a step function by a lowpass filter with idealized frequency cut-off. The Fourier transform yields in this case an output function starting at the time t→−∞ while the input function does not start until t=0. The reason for this phenomenon is that only mathematical concepts are used, while the causality law is a physical concept that has no meaning in pure mathematics. The causality law enters calculations typically through one of the partial differential equations of physics, their initial conditions, and their boundary conditions. No violation of the causality law will be encountered if the Fourier transform, and by implication other orthogonal transforms, are used for the solution of such problems with physical content.