Inverse problem for Orthogonal POlynomials

[zetafunction]

given a set of orthogonal polynomials

\int_{-\infty}^{\infty}dx P_{m} (x) P_{n} (x) w(x) = \delta _{m,n}

the measure is EVEN and positive, so all the polynomials will be even or odd

my question is if we suppose that for n-->oo

\frac{ P_{2n} (x)}{P_{2n}(0)}= f(x)

for a known function f(x) can we recover the measure ??

 

[Eynstone]

We must consider the possibility of different measures leading to the same f(x), as the limiting process for P_2n(x) doesn't involve the measure.

 

[Office_Shredder]

For example if you quadruple w(x), this results in each polynomial being halved. But halving P(x) and P(0) has no effect on f(x). So right off the bat you're stuck to getting a scaled measure at best.

 

[Mute]

If you know

\lim_{n \rightarrow \infty} \frac{P_{2n}(x)}{P_{2n}(0)} = f(x),
then do you not know the even polynomials themselves, up to scaling? Let P_{2n}(0) = 1 for simplicity. Then,

P_{2n}(x) = \sum_{k = 0}^n a_{2k} x^{2k},
so if we take the limit as n goes to infinity we have

\lim_{n \rightarrow \infty} P_{2n}(x) = \sum_{k=0}^\infty a_{2k} x^{2k} = f(x),
which basically is just a Taylor series for f(x). Hence we can identify

a_{2k} = \frac{1}{(2k)!} \left.\frac{d^{2k}}{dx^{2k}} f(x) \right|_{x = 0}.
Thus, since we know f(x), we know P_{2n}(x), at least in principle. (Finding a pattern for the derivatives may be difficult).

Since we know the even polynomials, we can in principle discover a differential equation which they solve. This will presumably turn out to be a Sturm-Liouville equation, which one can then use to find out the orthogonality condition, and hence the function w(x) (up to an overall scaling factor set by the choice of P_{2n}(0).

Anything obviously wrong with this procedure, in principle?