What non-zero function is w-orthogonal to all the polynomials of degree less than or equal to $n$?

Question

Background

It is understood that a function $q$ is $w$ orthogonal to a function $p$ over $[a,b]$ if there holds:

$$ \int_{a}^{b} q(x)w(x)p(x)dx = 0$$

For example, for $w(x):=1$, $[a,b]=[-1,1]$, the underlying $w$ orthogonal polynomials are the Legendre polynomials.

Question

Suppose now $w := \sqrt{1-x^2}$, and $p$ is any polynomial function of degree less than or equal to $n$, what is the non-zero function $q$ that is $w$ orthogonal to any $p$?

Answer 1

The Chebyshev polynomials of the second kind $U_n(x)$ are orthogonal with respect to $\sqrt{1-x^2}$ on $[-1,1]$.