Convergence analysis of Hermite approximations for analytic functions

In this paper, we present a rigorous analysis for root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and satisfy certain restrictions on the asymptotic behavior at infinity within this strip. The key ingredients of our analysis are some new and remarkable contour integral representations for the Hermite coefficients and the remainder of Hermite spectral interpolations with which sharp error estimates for Hermite approximations in the weighted and maximum norms are established. Further extensions to Gauss-Hermite quadrature and the scaling factor are also discussed. Particularly, we prove the root-exponential convergence of Gauss–Hermite quadrature under explicit conditions on the integrands. Numerical experiments confirm our theoretical results.