Orthonormal bases for $L^{2}(\mathbb{R})$ aside from Hermite functions?

Question

I was reading over Wikipedia and stumbled across the Hermite functions, which serve as an orthonormal basis for all real-valued square integrable functions on the entire real number line, using the inner product

$$\langle f, g \rangle = \int_{-\infty}^{\infty} f(x)g(x)dx$$

where $f$ and $g$ are always real-valued functions.

Unfortunately, the Hermite functions are a little too unwieldy and hard to work with for my purposes, given that they're generated from repeated differentiation; are there any other orthonormal bases for $L^{2}(\mathbb{R})$?

Answer 1

There are uncountably many choices for a basis for $L^2(\mathbb{R})$. Common choices include wavelet bases. They are given by $\psi_{j,k}(x)=2^{j/2}\psi(2^{j}x-k)$ where $j,k\in \mathbb{Z}$

Haar wavelets are a good introductory wavelet system to show this.