Some aspects of multidimensional convolution

It is shown that multidimensional convolution need not be associative. Further, for any positive integer k, it is shown that the multidimensional convolution of two real valued, bounded, integrable, nowhere zero functions defined in R^k can be identically equal to zero. These results are discussed in an algebraic setting, and a consequence involving random fields is briefly considered. Several aspects of multidimensional convolution that would be of interest to the signal processing community are included