is it true that for every set of orthogonal polynomials
$$ \int_{a}^{b} w(x)P_{n}(x)P_{m}(y) =a_{m,n}\delta _{m}^{n}$$
with respect a measure $ w(x)$ can we always find a quadrature formula for integrals in the form
$$ \int_{a}^{b}dx w(x)f(x) \approx \sum_{n} c_{n}f(x_{n}) $$
for the zeros of the orthogonal polynomials $ P_{n}(x_{n})=0 $
another question, let us supoose (in the sense of approximation theory that
$$ LIm_{n\to \infty} P_{2n}(x)= g(x) $$ for some function $ g(x) $
if this were true would be also true that the rooth of polynomials would tend to thre roots of $ g(x)$ ?? for big $ n\to 0 $