Analyticity of some kernels (Q1332680)

The \(m\)-th Hermite function is defined by \(H_ m(x)= h_ m(x) e^{-\pi x^ 2}\) where \(h_ m(x)\) is the \(m\)-th Hermite polynomial. The author studies the analyticity of the kernel \(\sum_{n\in N} \lambda_ n H_ n(x) H_ n(y)\) in relation to sequences \((\lambda_ n)_{n\in N}\). Using a detailed analysis of the Hermite functions, the author proves that the kernel is an analytic function for sequences for which there exist constants \(\rho>2\) and \(C>0\) with \(|\lambda_ n |\leq C/\rho^ n\), \(\forall n\in N\). This result is a partial answer to a question raised by A. L. Brown concerning the identity of sequences \((\lambda_ n)_{n\in N}\) for which the kernel is analytic. Questions of this kind arise in the study of eigenvalues of integral operators in relation to their kernels.