Matrix equation representation of the convolution equation and its unique solvability

Abstract

Abstract We consider the convolution equation F * X = B F* X=B , where F ∈ R 3 × 3 F\in {{\mathbb{R}}}^{3\times 3} and B ∈ R m × n B\in {{\mathbb{R}}}^{m\times n} are given and X ∈ R m × n X\in {{\mathbb{R}}}^{m\times n} is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and we analyze the unique solvability of the convolution equation.