# Polynomial upper approximation with respect to the Gaussian measure

Let $f = 1_{[a,+\infty)}$ be the indicator function of a half-line. Does there exist a sequence $(P_n)$ of polynomials such that $f(x) \leq P_n(x)$ for every real $x$ and $$\lim_{n\to \infty} \int_{\mathbf{R}} P_n d\gamma = \int_{\mathbf{R}} f d\gamma,$$ where $\gamma$ denotes the usual Gaussian measure ?

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have you tried Bernstein's polynomials of this function on [a-M,a+M] for large $M$ (maybe, somehow shifted)? –  Fedor Petrov Nov 30 '10 at 22:50
Yes, my impression was that this doesn't immediately work ... –  Guillaume Aubrun Dec 1 '10 at 9:14
I've been told offline that the relevant theorem to use is a weighted Weierstrass approximation theorem. Let $W(x)=\exp(x^2/2)$ ; given a continuous function $f$ so that $f(x)=o(W(x))$ at infinity, and $\epsilon >0$, there exists a polynomial $P$ so that $|P(x)-f(x)| \leq \epsilon W(x)$. This apparently is not so easy to prove (!?), but it is a standard problem to decide which functions $W$ satisfy this; a reference is: "Koosis, The logarithmic integral". Once you know this, few tricks suffice to prove what I asked (for example, positivity can be enforced by squaring the polynomial). –  Guillaume Aubrun Dec 1 '10 at 9:30