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Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function.

Consider $$ k_n(x,y)=\frac{h_{n+1}(x)h_n(y)-h_{n+1}(y)h_n(x)}{y-x}, $$ such that there exists a positive constant $\Gamma$ such that for any $n$ and for all $x$ $$ \int_{-\infty}^{-1}(k_n(x,y))^2dy< \frac{2\Gamma^2}{n+1}, \quad \int_{1}^{\infty}(k_n(x,y))^2dy< \frac{2\Gamma^2}{n+1}, $$ I am wondering if one can tell something about constant $\Gamma$, i.e. how small it can be?

Thank you for your help.

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+1 I guess your question is how small $\Gamma^2$ could be. Could you please provide some context to your question too, if possible. –  Sasha Jun 1 '12 at 0:10
    
Yes, you are right. How small it can be. I wanted to use Uspensky result jstor.org/discover/10.2307/… –  David Jun 1 '12 at 0:14
    
for the prolate spheroidal function instead of f(x) in the Uspensky theorem. –  David Jun 1 '12 at 0:15
    
Your $k_n(x,y)$ looks awfully like the Christoffel-Darboux expression for the Hermite polynomials... –  J. M. Jun 9 '12 at 14:25

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