Range of certain convolution operators and reconstruction from local averages

Abstract

For a compactly supported absolutely continuous measure $μ$ on ${\mathbb{R}}^2$ having a density function equal to a finite linear combination of indicator functions of rectangles $\left[a_{i}, b_{i}\right]\times \left[c_{i}, d_{i}\right],$ we analyse the range of the convolution operator $C_μ:C({\mathbb{R}}^2)\rightarrow C({\mathbb{R}}^2)$ defined by $C_μ(f)=f\starμ,$ where $(f\star μ)(x,y)=\int_{\mathbb{R}^2}f(x-s,y-t)dμ.$ It is shown that $C_μ$ maps the space of all continuous functions $C({\mathbb{R}}^2)$ onto the space $C^{2*}({\mathbb{R}}^2)=\{f:{\mathbb{R}}^2\rightarrow {\mathbb{C}}:\frac{\partial^2 f}{\partial x \partial y},\frac{\partial^2 f}{\partial y \partial x}\in C({\mathbb{R}}^2)\}$ provided the density function of $μ$ satisfies certain conditions.