# Polynomials are dense in weighted $L^2$ space

Hi,

It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight $x^{\alpha-1}e^{-x}/\Gamma(\alpha)\;dx$.

Is there any reference to this fact preferrably including the condition which property of the weight implies the density of polynomials?

My guess is that if $$\int_{-\infty}^{+\infty}e^{-\lambda|x|}P(dx)<\infty$$ for some $\lambda>0$, then the polynomials are dense in $L^2(\mathbb{R},P)$.

-

Sure, the beautiful book by N.I.Akhiezer The Classical Moment Problem and Some Related Questions in Analysis, where in particular you can find a thorough discussion of the property of the density of the polynomials in $L^1$ and $L^2$ for measures with finite moments of all order (together with sufficient conditions and counter-examples to the density.)

-
Did you mean Akhiezer alone, "Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd." ? –  Leonid Petrov Sep 30 '11 at 16:19
Anyway, the book "Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd." is the required reference, and it contains an original reference to Riesz, "Sur le probleme des moments et le theoreme de Parseval correspondant, Acta Litt. ac Sci.t Szeged. 1 (1922), 209-225." Thank you! –  Leonid Petrov Sep 30 '11 at 16:27
Yes you're completely right, sorry, I meant Akhiezer alone. Corrected. –  Pietro Majer Sep 30 '11 at 16:36