Timeline for Completeness of discrete shifts in $\mathbb{R}^n$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 4, 2019 at 21:53 | comment | added | Muzi | One result which is somehow designed for Gaussians is due to Zalik, researchgate.net/publication/… , Theorem 2 | |
| Dec 4, 2019 at 21:15 | comment | added | Bunyamin Sari | What is the theorem that everyone seems to be referring here? Given a function (with an assumption) how do you get the sequence $(x_n)$ so that $f(\cdot-x_n$'s are dense in $L_2$? I am aware of Olevskii's theorem (sciencedirect.com/science/article/pii/S0764444297878731) but you seem to be referring to something else. | |
| Dec 4, 2019 at 19:47 | comment | added | Pierre PC | 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. | |
| Dec 4, 2019 at 18:21 | comment | added | Sam Zbarsky | 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. | |
| Dec 4, 2019 at 17:26 | comment | added | Muzi | Thank you for the comment. How does the density of functions with smooth and compactly supported Fourier transform helps me here? | |
| Dec 4, 2019 at 11:55 | history | edited | Muzi | CC BY-SA 4.0 |
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| Dec 3, 2019 at 16:34 | history | edited | Willie Wong |
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| Dec 3, 2019 at 15:36 | comment | added | Sam Zbarsky | 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. | |
| Dec 3, 2019 at 14:30 | review | First posts | |||
| Dec 3, 2019 at 15:20 | |||||
| Dec 3, 2019 at 14:25 | history | asked | Muzi | CC BY-SA 4.0 |