I need to construct a sequence of orthogonal polynomial $(P_i)_{i=0}^{\infty}$ for a family of scalarproducts. I want to look at different scalar products $\langle P_n(x),P_m(x) \rangle_{\mu(x)}=\int_a^b P_n(x)P_m(x) \, d\mu(x)$ with differing intervals $[a,b]$ and differing measures $\mu(x)$.
I know that for certain choices these polynomials are known, yielding i.e. the Laguerre Polynomials. Actually there seems to be a wealth of different orthogonal polynomials out there, making it hard for me to grasp the relevant concepts. Is there a construction method for obtaining these orthogonal polynomials? Is this always possible or only under certain conditions? Does anyone have an idea how to somewhat systematically understand why different measures lead to certain families of orthogonal polynomials?
You can apply the Gram–Schmidt process to the sequence $1,x,x^2,x^3,\ldots$ if you can evaluate the integrals.