Range of certain convolution operators and reconstruction from local averages

TL;DR

This paper characterizes the range of a convolution operator with measures supported on rectangles in the plane, showing it maps continuous functions onto a space with continuous mixed second derivatives under certain conditions.

Contribution

It provides a detailed analysis of the range of convolution operators with measures supported on rectangles, extending understanding of their functional mapping properties.

Findings

Convolution operator maps C(R^2) onto functions with continuous mixed second derivatives.

Range characterization depends on conditions on the measure's density function.

Results applicable to reconstruction problems 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 $C_{\mu}:C({\mathbb{R}}^2)\rightarrow C({\mathbb{R}}^2)$ 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 $C^{2*}({\mathbb{R}}^2)=\{f:{\mathbb{R}}^2\rightarrow {\mathbb{C}}:\frac{\partial^2 f}{\partial x \partial y},\frac{\partial^2 f}{\partial y \partial x}\in C({\mathbb{R}}^2)\}$ provided the density function of $\mu$ satisfies certain conditions.