# Completeness of discrete shifts in $\mathbb{R}^n$

Consider the space $$L^2(\mathbb{R})$$. Let $$(x_n)_n \subset \mathbb{R}$$ be a sequence and $$f \in L^2(\mathbb{R})$$ a functions. It is well understood under which assumptions the span of the set $$S = \{ f(\cdot - x_n) : n \in \mathbb{N} \}$$ is dense in $$L^2(\mathbb{R})$$. This goes into the theory of generators of $$L^2(\mathbb{R})$$. In particular, one can show that if $$f$$ is a Gaussian then we can find a sequence $$(x_n)_n$$ such that the span of $$S$$ is dense.

Can this be generalized to $$L^2(\mathbb{R}^d)$$ where $$d > 1$$? In this case, $$(x_n)_n$$ would be a sequence in $$\mathbb{R}^d$$. Is it again possible to choose $$f$$ to be a Gaussian? So far, I didn't find results on this question in the literature.

• Fourier transform everything and use density in $L^2$ of functions whose Fourier transform is $C_c^\infty$. It shouldn't be hard to prove then. – Sam Zbarsky Dec 3 '19 at 15:36
• Thank you for the comment. How does the density of functions with smooth and compactly supported Fourier transform helps me here? – Muzi Dec 4 '19 at 17:26
• If you fourier transform, the shifts of a Gaussian become multiplications of a Gaussian by functions $\exp(i\xi\cdot x)$. By density, we only need to approximate $C_c^\infty$ functions. Let $g$ be such a function, and say that we are trying to approximate it within $\epsilon$ in $L^2$. Take a big box of large side length $R$, on that box use Fourier series to approximate $g/f$ well. Hopefully your errors outside of the box will be small, since $f$ decays very quickly. Obviously it will take some work to bound all the errors. – Sam Zbarsky Dec 4 '19 at 18:21
• If you only need one sequence, let me finish the argument for people who don't like to bound errors. I choose the image of the sequence to be all of $\mathbb Z^d$. The density of $C^\infty_c$ shows that functions of the form $gf$ for $g$ bounded, smooth, $(k\mathbb Z)^d$-periodic for some $k\in\mathbb N$, are dense; indeed, for $h\in C^\infty_c$, take the $2k$-periodic function $g_k$ that coincides with $g/f$ on $[-k,k]^d$. Such periodic $g$ are uniform limits of trigonometric polynomials (usual Fourier series), so $gf$ are $L^2$-limits of the set of functions described by Sam. – Pierre PC Dec 4 '19 at 19:47
• One result which is somehow designed for Gaussians is due to Zalik, researchgate.net/publication/… , Theorem 2 – Muzi Dec 4 '19 at 21:53