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Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any $\epsilon>0$. As a consequence, low-frequency trigonometric functions are dense in $L^2([a,b])$. The proof of the first result uses Hermite functions, and the second one follows by taking the Fourier transform of the first. (Arxiv link.)

Two reviewers told us that these results "must be known" but didn't provide references.

Two questions:

  • Are these results folklore?
  • What's a good place to look for similar/related results?
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While I don't have an answer to your question: any time one sees a family having dense linear span in a function space, it's natural to wonder if this isn't a case of "approximation by convolution with a suitable kernel". E.g. trig polys are dense in C(T) - convolve with Fejer kernel. Of course the striking thing in your result is that one doesn't need to dilate/shrink the Gaussians, so I'm not sure if what I've said is all that relevant –  Yemon Choi Dec 16 '09 at 21:00

3 Answers 3

up vote 2 down vote accepted

Wiener's approximation theorem says that

Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{R}\}$ is dense in $L^2(\mathbb{R})$ if and only if zeros of the Fourier transform of $h$ has zero Lebesgue measure.

See Wiener's book "The Fourier Integral and Certain of Its Applications" or Chandrasekharan's "Classical Fourier Transforms".

Further question when the translating parameters are restricted to a smaller set are considered by a series of authors. See this paper for instance and the reference therein.

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The reference to Faxén's paper was exactly what we were looking for. Thanks! –  Axel Boldt Feb 9 '10 at 16:48

I think that the results are "folklore" in the sense that if you have no concern at all for numerical stability, using small translates of the Gaussian to obtain derivatives is the first thing that you'd think to do. Likewise with the trig functions with frequency $\omega \ll 1$, you can just directly differentiate many times with respect to $\omega$ and then evaluate at $0$. You get polynomials, which are dense by the Weierstrass approximation theorem. You say several times in the paper that the results are surprising, but in my opinion the qualitative results are not all that surprising.

These results are at least similar to popular results that certain sets of wavelets are bases, and certain other sets span but are overcomplete. I don't have a reference, I just remember hearing about a set of wavelets on $\mathbb{R}$ that comes from a general lattice in $\mathbb{C}$. Whether the wavelets span or are overcomplete depends on the determinant of the lattice. This is not the same result, but it is similar.

Your paper also analyzes the numerical stability of your approximations. That seems like the more original contribution to me. Maybe the referees would be happier if you shifted the emphasis of the paper to the numerical stability issue.

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Wiener's tauberian theorems seem relevant to your results (in fact this has been discussed recently on mathoverflow here: Is there an L^p tauberian theorem?). One of these theorems states that the linear combinations of the translates of a function, f, is dense in L^2(R) if and only if the Fourier transform of that function doesn't vanish on a set of positive measure. Since the Fourier transform of the Gaussian is a Gaussian (and doesn't vanish) this is very similar to your Theorem 1. Of course, Wiener's theorem only gives you an approximation of the form $ \sum_{n=1}^{N} a_{n} e( (x- t_n)^2)$ for some sequence of real numbers $t_{n}$. You might be able to recover your result by approximating the the sequence $t_{n}$ by a sequence of the for m $\{nt\}$ (after possibly adjusting the coefficients a_n), however this isn't immediate to me.

If these remarks sound along the lines of what you are looking for, Wiener's book "the Fourier integral and certain of its applications" is a good reference. Jacob Korevaar's Tauberian Theory: A Century of Developments has a much more comprehensive (and current) treatment of the subject.

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