If one has a function $h(x)$ such that only some of its moments are finite (the rest are infinite):
$\int x^k h(x) \mathrm{d} x<\infty$ for $k<N$
e.g. A rational function ($h(x)=\frac{a}{x^2+b^2}$).
When is it possible to define a set of orthonormal functions $\{f_n(x)\}_{n=1}^{\infty}$ such that:
1. $\int h(x) f_n(x) \mathrm{d}x=0$
2. $\int f_m(x) f_n(x) \mathrm{d}x=\delta_{nm}$
3. $\int x f_m(x) f_n(x) \mathrm{d}x=\delta_{nm} \int x f_n(x)^2 \mathrm{d}x<\infty$
4. The set including $h(x)$ and all the functions $\{f_n(x)\}_{n=1}^{\infty}$ is complete for $L^2(\mathbb{R})$ (or at least a complete as the space of all polynomials).
The integral could be infinite $(-\infty,\infty)$ or semi-infinite $(0,\infty)$ (I am interested in both cases).
Further details: I do not require a form for the $f_n(x)$ I just need to know for which $h(x)$ they exist.
Progress:
1. I have a solution when all the moments of $h(x)$ are finite using two sets of orthonormal polynomials.
2. If $h(x)$ is odd and the integration range is $(-\infty,\infty)$ I can satisfy all the requirements except completeness (4) by setting $f_n(x) = P_{2n}(x)e^{-x^2}$ where $P_{n}(x)$ is an orthonormal polynomial defined with respect to the measure $e^{-2x^2}$:
that is $\int_{-\infty}^{\infty} P_n(x) P_m(x) e^{-2x^2} \mathrm{d}x=\delta_{nm}$.
3. I have considered the Cayley transform ($y=\frac{x-1}{x+1}$) for the integral with range ($(0,\infty)$) to map the range to $(-1,1)$ but as yet without success.
Context: I am attempting to map between two Hamiltonians and can do so if these relations are satisfied.