The pointwise product in Weyl quantization

We study the circle dot-product of Bracken [1], which is the Weyl quantized version of the pointwise product of functions in phase space. We prove that it is not compatible with the algebras of finite rank and Hilbert-Schmidt operators. By solving the linearization problem for the special Hermite functions, we are able to express the circle dot-product in terms of the component operators, mediated by the linearization coefficients. This is applied to finite rank operators and their matrices, and operators whose symbols are radial and angular distributions.