Differentiate and make waves (Q1912360)

How derivatives of functions and finite differences of functions are related globally to each other? The paper is devoted to the investigation of this problem in the context of making waves and wavelets by using these operations. We note that continuous wavelet transforms can be built by making use of ``wavelet measures'' rather than wavelet functions. These measures should have a prescribed number of vanishing moments. In particular, finite differences can be included into this scheme. The choice of wavelet measures is very large. This point of view was developed in the recent book of the reviewer , ``Fractional integrals and potentials'' [Harlow: Addison Wesley Longman Ltd. (1996)].