Asymptotic Analysis of Even Hermite–Sobolev Orthogonal Systems

This poster presents an analysis of an even Hermite–Sobolev system associated with the Gaussian weight rho(x) = exp(−x²) on the real line. The considered system is generated from normalized even Hermite functions and naturally arises in functional analysis, spectral theory, and the asymptotic study of generalized orthogonal systems.

Let r ≥ 1 be a fixed integer. The family of functions is defined by polynomial expressions for the initial indices and by integral representations involving even Hermite functions for higher orders. A fundamental property of this construction is that the r-th derivative of the system coincides with the classical even Hermite functions. We first prove that this family forms an orthogonal system with respect to a Sobolev-type inner product restricted to the space of even functions. This orthogonality reflects the intrinsic differential structure of the system and highlights its close connection with Hermite analysis.

The main objective of this work is to investigate the asymptotic behavior of the system as the index tends to infinity. By combining the classical Plancherel–Rotach asymptotic formulas for Hermite functions with the steepest descent method applied to the associated integral representations, we identify three distinct asymptotic regimes: the oscillatory (bulk) region, the transition region, and the exponential (outer) region. In particular, we show that the asymptotic behavior in the transition region is governed by the Airy function, revealing a universal feature characteristic of Gaussian-based orthogonal systems. Finally, numerical illustrations are provided to demonstrate the consistency between the asymptotic results and computational observations.