Infinite Matrix Representations of Isotropic Pseudodifferential Operators

Abstract

We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of order r if and only if its "matrix" $(K(A))_{m,n} := < A\phi_n,\phi_m>_{L^2(\mathbb{R})}$ is rapidly decreasing away from the diagonal $\{m = n\}$, order $\frac {r}{2}$ in $m + n$, and where applying the discrete difference operator along the diagonal decreases the order by one. Additionally, we use this result to prove an analogue of Beal's theorem for isotropic pseudodifferential operators.