# Construct an orthonormal basis

Consider vector space of functions continuous on $I \subseteq \mathbb R$ and scalar product is defined as $({\bf f}, {\bf g}) \equiv \int_{I} \rho(x) f(x)g(x) dx$. Let the generating function $G(z,x)$ be known. Let the span of $\phi_n(z) = \frac{\partial^n}{\partial x^n}\left. G(z,x)\right|_{x=0}$ be the subspace. How to construct orthonormal basis of this subspace?

Ok, obvious way is to use the Gram-Schmidt procedure. But I would like to utilize the generating function in a manner similar to that in Krylov subspace I can use Lanczos technique.

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What do you know about $G(z,x)$? –  Christopher A. Wong Feb 1 '13 at 21:53
An exact expression. –  0x2207 Feb 2 '13 at 6:44