Timeline for Polynomial upper approximation with respect to the Gaussian measure
Current License: CC BY-SA 2.5
5 events
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| Dec 1, 2010 at 9:30 | comment | added | Guillaume Aubrun | 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). | |
| Dec 1, 2010 at 9:14 | comment | added | Guillaume Aubrun | Yes, my impression was that this doesn't immediately work ... | |
| Nov 30, 2010 at 22:50 | comment | added | Fedor Petrov | have you tried Bernstein's polynomials of this function on [a-M,a+M] for large $M$ (maybe, somehow shifted)? | |
| Nov 29, 2010 at 10:31 | history | edited | Guillaume Aubrun |
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| Nov 29, 2010 at 9:46 | history | asked | Guillaume Aubrun | CC BY-SA 2.5 |