My thesis is all about classical orthogonal polynomials. Orthogonal polynomials are defined to be a sequence of polynomials whose inner product is zero. That is, the integral $$\int_a^b f(x)g(x)w(x)\:dx =0$$ where $w(x)$ is the nonnegative weighting function.
Now, my problem is what is the general formula in obtaining the weight function? Is it given already for a certain family of orthogonal polynomials or it is derived using a specific formula?
In the general context, the weight function $w(x)$ will be prescribed on the interval (a,b). Then the task is to find the orthogonal polynomial sequence ${p_n(x) }$ . Refer to text like "An introduction to Orthogonal Polynomials " by Theodore S. Chihara to get further details.
On the contrary,supposing if, a polynomial sequence ${p_n(x)}$ is prescribed on the interval (a,b) as orthogonal. It will be incomplete without the definition of the inner product. In which case, the function $ w(x) $ need also be prescribed with such definition.