Range of certain convolution operators and reconstruction from local averages

Abstract

For a compactly supported absolutely continuous measure $\mu$ 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 $C_{\mu }:C({\mathbb{R}}^2)\to C({\mathbb{R}}^2)$ defined by $C_{\mu }(f)=f\star \mu ,$ where $(f\star \mu )(x,y)={\int }_{{\mathbb{R}}^2}f(x-s,y-t)d\mu .$ It is shown that $C_{\mu }$ maps the space of all continuous functions $C({\mathbb{R}}^2)$ onto the space $C^{2\ast }({\mathbb{R}}^2)=\{f:{\mathbb{R}}^2\to \mathbb{C}:\frac{{\partial }^{2}f}{\partial x\partial y},\frac{{\partial }^{2}f}{\partial y\partial x}\in C({\mathbb{R}}^2)\}$ provided the density function of $\mu$ satisfies certain conditions.

18 pages