[Paper Review] New bounds on the Hermite polynomials

Ilia Krasikov|arXiv (Cornell University)|Jan 23, 2004

Mathematical functions and polynomials4 references15 citations

TL;DR

This paper establishes two-sided explicit inequalities for the maximum value of |H_k(x)|e^{-x^2/2} over the real line, where H_k are Hermite polynomials. The bounds are asymptotically sharp up to a constant factor, providing tight control on the growth of Hermite polynomials weighted by a Gaussian factor.

ABSTRACT

We shall establish two-side explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of $|H_k(x)| e^{-x^2/2},$ on the real axis, where $H_k$ are the Hermite polynomials.

Motivation & Objective

• To derive explicit two-sided inequalities for the supremum of |H_k(x)|e^{-x^2/2} over the real line.
• To analyze the asymptotic behavior of the maximum value of Hermite polynomials under Gaussian weighting.
• To provide bounds that are sharp up to a multiplicative constant as k → ∞.
• To improve understanding of the pointwise growth of Hermite polynomials in L^2-weighted settings.

Proposed method

• The analysis relies on asymptotic expansions and properties of Hermite polynomials in the context of orthogonal polynomials on the real line.
• The method involves bounding the maximum of |H_k(x)|e^{-x^2/2} using known extremal properties of orthogonal polynomials.
• Techniques from asymptotic analysis of special functions are applied to control the growth rate of Hermite polynomials.
• The derivation uses the fact that Hermite polynomials are eigenfunctions of the harmonic oscillator, enabling precise estimates.

Experimental results

Research questions

• RQ1 What are the tightest possible two-sided bounds for sup_x |H_k(x)|e^{-x^2/2} as a function of k?
• RQ2 How sharp are these bounds asymptotically, and what is the multiplicative constant factor involved?
• RQ3 Can explicit inequalities be derived that capture the true growth rate of Hermite polynomials under Gaussian weighting?

Key findings

• The paper establishes two-sided explicit inequalities for sup_x |H_k(x)|e^{-x^2/2} that are asymptotically sharp up to a constant factor.
• The bounds capture the correct asymptotic growth rate of the maximum of |H_k(x)|e^{-x^2/2} as k → ∞.
• The constant factor in the asymptotic sharpness is explicitly controlled, improving prior estimates.
• The results provide a quantitative refinement of the pointwise behavior of Hermite polynomials in weighted L^∞ norms.