Abstract
We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of a system of partial difference operators. We show that whenever the space of homogeneous solutions of a system of partial difference operators is finite dimensional, it always has to consist of exponential polynomials which can be described exactly. We give an explicit relationship between the polynomial part of theses spaces in a direct though somewhat intricate relation to the multiplicity of the common zeros of certain multivariate polynomials. This concept was introduced by Grobner in his characterization of the kernels of partial differential operators with constant coefficients. Based on the characterization of the kernels of convolutions, we can then also determine the kernels of stationary subdivision operators based on arbitrary scaling matrices.
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