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

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 $L^{2}(\mathbb{R})$?

• What are your purposes and why are Hermite functions not suited for them? Wavelet bases are now very popular and comfortable to work with. May 6, 2017 at 3:15
• That scalar product would have no chance for convergence for polynomials. You need a metric (window) to make it. For example $e^{-x^2}$. May 6, 2017 at 4:05
• @guest In an ideal world, I'd like a basis that's easy to integrate over and where each basis vector doesn't needed to be generated iteratively. However, I'm more interested to see a broad list of bases to choose from; the more options the better. May 6, 2017 at 4:21
• @mathreadler I was referring to Hermite functions, not Hermite polynomials. The former converge, the latter don't. At least, I'm pretty sure; I'm rather new to this area of mathematics. May 6, 2017 at 4:24
• Yes, ok that makes sense. May 6, 2017 at 5:05

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 $j,k\in \mathbb{Z}$
• Thank you! Just one point I need to clear up: I checked the Wikipedia page for Haar wavelets, and it seems like your definition is a teeny bit different from yours. They use $\psi_{j,k}= 2^{j/2} \psi(2^{j}x-k)$, while you use $\psi_{j,k}= 2^{j/2} (2^{j}x-k)$. Is this a typo on your part, or are you referencing something different from Wikipedia? May 6, 2017 at 4:17