orthogonal polynomials and weight functions

Question

how are orthogonal polynomials related to their weight function? is there an algebraic relationship other than the defining integral $$\int_a^b w(x)P_n(x)P_m(x)\,dx$$?

thanks for the help!

Answer 1

To say that the polynomials are orthogonal implicitly references the inner product $$\langle f, g \rangle = \int_a^b f(x)g(x)w(x)dx$$ The closest thing I can think of to an algebraic relationship between the polynomials and the weight function is the requirement that $$\langle P_n, P_m \rangle = \delta_{nm}$$

Answer 2

If one is given a second order linear differential operator of the form $$L(f) \equiv a(x)f''(x)+b(x)f'(x)+c(x)f(x)$$ and one wishes to know for which Hilbert space $L$ constitutes a Hermitian operator, one puts $L$ into the form known as Sturmian form $$L(f) \equiv \frac{1}{r(x)}\big((p(x)f'(x))'+q(x)f(x)\big)$$ where it is simple to show that $$p(x)=e^{\int \frac{b(x)}{a(x)}}$$ $$r(x)=\frac{p(x)}{a(x)}$$ and $$q(x)=r(x)c(x)$$

Then $r(x)$ is the natural weighting function to use in order to turn $L$ into a Hermitian operator (one still needs to add boundary constraints) and the eigenvalue solutions to $L$ will form an orthogonal basis for the constructed Hilbert space. See Holland's 'Applied Analysis by the Hilbert Space Method' for details and proofs. Good question!