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

  • $\begingroup$ What are your purposes and why are Hermite functions not suited for them? Wavelet bases are now very popular and comfortable to work with. $\endgroup$
    – guest
    May 6, 2017 at 3:15
  • $\begingroup$ That scalar product would have no chance for convergence for polynomials. You need a metric (window) to make it. For example $e^{-x^2}$. $\endgroup$ May 6, 2017 at 4:05
  • $\begingroup$ @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. $\endgroup$
    – SilasLock
    May 6, 2017 at 4:21
  • 1
    $\begingroup$ @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. $\endgroup$
    – SilasLock
    May 6, 2017 at 4:24
  • $\begingroup$ Yes, ok that makes sense. $\endgroup$ May 6, 2017 at 5:05

1 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.

  • $\begingroup$ 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? $\endgroup$
    – SilasLock
    May 6, 2017 at 4:17
  • $\begingroup$ yes it was a typo, now fixed! $\endgroup$ May 6, 2017 at 4:25
  • $\begingroup$ Awesome, thanks! $\endgroup$
    – SilasLock
    May 6, 2017 at 4:25

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