We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ 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 analyze the unique solvability of the convolution equation.

翻译：考虑卷积方程$F*X=B$，其中$F\in\mathbb{R}^{3\times 3}$与$B\in\mathbb{R}^{m\times n}$为已知矩阵，$X\in\mathbb{R}^{m\times n}$为待求矩阵。该卷积方程可视为具有特殊结构系数矩阵的线性系统，这一性质催生了包括高效数值算法在内的诸多研究。本文证明卷积方程可表示为广义Sylvester方程，并针对图像处理中的实际算例，展示该广义Sylvester方程可简化为更简洁的形式，进而分析卷积方程的唯一可解性。