Talk:Babenko–Beckner inequality

Lack of rigor, hand-waving, or maybe I just don't get it.

How do the elementary symmetric polynomials converge to the Hermite polynomials? It's too deep for me. Deepmath (talk) 00:53, 2 August 2009 (UTC)

First, I would like to point out that he does not claim what you have said. It is very subtly, but very importantly, different. He shows a relationship between the elementary symmetric polynomials of n variables and the Hermite polynomial of the sum of those n variables where in the case of the elementary symmetric polynomials we only consider the points that the measure we are considering is non-zero. In Beckner's paper, Inequalities in Fourier Analysis (Annals of Mathematics, 102 (1975), 159-182), he explains the relationship between the elementary symmetric polynomials and the Hermite polynomials by comparing their generating function. This doesn't actually prove what he next states, the relationship between them, but it goes a long way to convincing us that a close relationship exists. Now, the actual proof of his claim happens in the Appendix. First he notes that for a fixed n, the first 3 elementary symmetric polynomials and the Hermite polynomials in fact have exactly the relationship shown. It is very easy to compute this directly. Next, he claims two iterative relationships (one for the elementary symmetric and one for the Hermite polynomials) that are very similar; he proves these using some basic calculus, algebra and some combinatorial arguments on both the generating functions and the explicit definitions. Now, he compares the iterative relationships and these show that the claim on the relationship between the elementary symmetric polynomials is given by equality with an error term. This relationship is equation (5) in his paper and the error term for a fixed polynomial l (as in the lth Hermite polynomial and the lth elementary symmetric polynomial) tends to zero as n tends to infinity. Thus, when he takes the limit of his measures dν_n to obtain a relationship in dμ, the difference between the elementary symmetric polynomials and the Hermite polynomials tends to zero. Grimtageuk (talk) 18:45, 12 May 2013 (UTC)