On certain integral operators associated with Hermite and Laguerre polynomials
Motivated by applications in nonparametric curve estimation theory, we consider orthogonal polynomial representations for approximation of functions. In particular, we look at the classical Hermite and Laguerre polynomials that have been used in nonparametric statistics where instead of the complete knowledge of a function only its noisy samples are available. In order to study the consistency of the corresponding nonparametric series function estimators one should examine the first, second and often higher moments of estimators. These evaluations are related to higher powers of the associated kernel function of the selected approximating expansion. Hence, in this paper we investigate the asymptotic behavior of kernel operators related to powers of kernels resulting from such orthogonal polynomial expansions. We examine the case of kernel operators associated with Hermite and Laguerre polynomials.