Chromatic derivatives and non-separable inner product spaces

We present a detailed motivation for the notions of the chromatic derivatives and the chromatic expansions. The chromatic derivatives are special, numerically robust linear differential operators. The chromatic expansions are the associated local expansions, which possess features of both the Taylor and the Nyquist expansions. We give simplified treatment of some of the basic properties of the chromatic derivatives and the chromatic expansions. We then introduce some new spaces of signals with a scalar product defined by a Cesàro–type sum of products of chromatic derivatives. In one such space any two sine waves of different positive frequencies are orthogonal. We finally consider an approximation of such scalar product that is relevant for signal processing.