# inverse problem with Orthogonal polynomials

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
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
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