A limitation of the kernel method for joint distributions of arbitrary variables

Recently, Cohen has proposed a construction for joint distributions of arbitrary physical quantities, in direct generalization of joint time-frequency representations. Actually, this method encompasses two approaches: one based on operator correspondences and one based on weighting kernels. The literature has emphasized the kernel method due to its ease of analysis; however, its simplicity comes at a price. In this letter, we use a simple example to demonstrate that the kernel method cannot generate all possible bilinear joint distributions. Our results suggest that the relationship between the operator method and the kernel method merits closer scrutiny.