Examples of orthogonal/orthonormal functions which are not finite degree polynomials?

Question

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.

Answer 1

Start with any continuous differential positive function $p$ on $[0,1]$, any integrable function $q$, and a continuous positive weight function $w$. Let $\alpha,\beta \in [0,\pi)$ be given. Then there are infinitely many real values of $\lambda$, say $\lambda_{0} < \lambda_{1} < \lambda_{2} < \cdots$, for which non-trivial solutions of the following problem exist: $$ -(pf')'+q = \lambda w f,\\ \cos\alpha f(0)+\sin\alpha f'(0)=0,\\ \cos\beta f(1)=\sin\beta f'(1) = 0. $$ These solutions $\{ f_{n} \}_{n=0}^{\infty}$, when normalized so that $\int_{0}^{1}f_{n}^{2}w dx=1$, form a complete orthonormal basis of the weighted space $L^{2}_{w}[0,1]$ whose inner-product is $(f,g)_{w}=\int_{0}^{1}fgwdx$.

Answer 2

Check this set

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

with inner product defined as

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

Answer 3

I think the most common one is: $(e^{inx})_{n \in \mathbb{N}}$, with inner product:

Letting $e_n(x) := e^{inx}$,

$<f, e_n>=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx} dx$.

$e^{-inx}$ means complex conjugation, and we integrate these because $\sum$ becomes $\int$ when continuous. $\frac{1}{2\pi}$ is for normalization.

Actually, this is the same as Benghorbal's answer.