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 ( f ⋆ μ ) ( x ) = ∫ R f ( x − y ) d μ ( y ) , 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 C μ ( f ) = f ⋆ μ maps the space of all r times continuously differentiable functions C r ( R ) onto the space of all r + 1 times continuously differentiable functions C r + 1 ( R ) provided the density function of μ is not equal to zero almost everywhere.