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We present a detailed motivation for the notions of chromatic derivatives and chromatic expansions. Chromatic derivatives are special, numerically robust linear differential operators; chromatic expansions are the associated local expansions, which possess the best features of both the Taylor and the Nyquist expansions. We give a simplified treatment of some of the basic properties of chromatic derivatives and chromatic expansions which are relevant for applications. We also consider some signal spaces with a scalar product defined by a Cesaro-type sum of products of chromatic derivatives, as well as an approximation of such a scalar product which is relevant for signal processing. We also introduce a new kind of local approximations based on trigonometric functions.