On compositional and convolutional discrete systems

During the last several years many publications concerning two and multidimensional digital filters have appeared (see [1], [2] and [3]). Having presented six counterexamples E. I. Jury [4] has shown that in extending the theorems developed for the one dimensional case to two and multidimensional systems many difficulties are encountered. It seems to be useful to investigate the relationship among fundamental concepts of multidimensional digital filters. In this paper we shall consider the following crucial notions of discrete system theory in the multidimensional time area: linearity, causality, time independence and stability. We shall use standard notation and terminology of linear space theory: linear spaces and subspaces, bases, and linear mappings.