The Inversion of Multiscale Convolution Approximation and Average of Distributions

Abstract

We develop two kinds of inversion formulas of the multiscale convolution approximation which is defined by a convolution kernel φ. The inversion formulas are constructed by a convolution kernel ψ which is defined in terms of φ and has a vanishing moment of order one. A large class of generalized moving average approximations with B-splines, Box-splines and exponential Box-splines (EB-splines) as convolution kernels φ is included in the theory formulated in this paper. The average of distributions is considered, and correspondingly, the formulas related to the EB-splines are obtained from the δ-average.