<title>Decomposition and inversion of von Neumann-like convolution operators</title>

Abstract

Methods are presented for the decomposition of two-dimensional von Neumann-like convolution operators into sums and products of smaller von Neumann-like operators. Consequences of the techniques include the face that ever second-order operator is the sum and product of five or fewer first-order operators. A totally symmetric second-order operator can be written as the sum and product of three or fewer first-order operators. In general, an nth-order von Neumann-like operator can always be written as the sum and product of three lower order operators. The following inversion result is also discussed. If the (circulant) von Neumann mean filter operator is defined on a square coordinate set with n rows and n columns, then it fails to be invertible only if either the integer 5 divides n or the integer 6 divides n. This result provides a partial solution to a question posed by P. Gader.