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Let $f(x)$ be the function which can be considered the limit $f(x)= \lim_{n \to \infty} C_n P_{n} (x)$, where $P_{n}(x) $ is the $n$-th orthogonal polynomial with respect to a positive even measure density, i.e. $w(x) >0 $ and $ w(x)=w(-x) $.

Additionally, the moment problem $\mu _{n} = \int _{-\infty}^{\infty}x^{n}w(x)\mathrm{d}x$ is determined.

Then my question is, if I know $f(x) $ can I get the set of orthogonal polynomials?

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Your question's confusing. You have a $p$, and then a $P$; make up your mind! Is your $w(x)$ known or unknown? –  J. M. Nov 22 '11 at 9:53
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The limit doesn't make sense to me. –  Hans Lundmark Nov 22 '11 at 9:57
    
the limit $ n \to \infty $ means that for high degree the orthogonal polynomial will tend to a given function $ f(x) $ the measure $ w(x) $ is not known :( –  Jose Garcia Nov 22 '11 at 12:41
    
@JoseGarcia I edited to you post. Please see if did not inadvertently modify the meaning. Using words in capitals is equivalent to shouting on the net. I replaced that with a different styling. –  Sasha Nov 22 '11 at 14:11
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Can you give an example for $C_n$ and $f(x)$ for some simple family of orthogonal polynomials? –  Phira Nov 22 '11 at 22:39

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