Reconstruction of continuous functions from locally uniform weighted averages

The aim of the paper is to solve the convolution equation of the type f⋆μ=g, where f⋆μ is the convolution of f and μ defined by , g is a given function and μ is a compactly supported absolutely continuous measure with a density function equal to a finite linear combination of indicator functions of intervals. It is shown that the operator Cμ(f)=f⋆μ maps the space of all r times continuously differentiable functions Cr(R) onto the space of all r+1 times continuously differentiable functions Cr+1(R) provided the density function of μ is not equal to zero almost everywhere.