Examples of orthogonal/orthonormal functions which are not finite degree polynomials? I've been reading "Fourier Series & Orthogonal Polynomials" by Dunham Jackson. Great introductory read for anyone interested by the way!
My question is, what are other examples of Orthogonal Functions, with respect to a weight function if necessary, which are not finite degree polynomials?
Jackson's book gives the Bessel functions as one example, e.g.
$$\int_0^1 xJ_0(\lambda x)J_0(\mu x)dx = 0.$$
It also discusses the $\sin $ and $\cos $ functions of course.
The other examples are finite degree polynomials including Legendre, Jacobi, Hermite, and Laguerre polynomials.

Check this set 

$$ \left\{ 1,\sin(x),\cos(x),\sin(2x),\cos(2x),\dots \right\} $$

with inner product defined as

$$\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) g(x) dx. $$

