We consider the set $C_0(\mathbb R)$ of real continous functions $f:\mathbb R\rightarrow \mathbb R$ with $\lim_{|x|\rightarrow \infty}f(x)=0$ endowed with the supremum norm. Is there $f\in C_0(\mathbb R)$ such that the linear span of $f(2^k \, \cdot \,)$, $k\in \mathbb N$, is dense in $C_0(\mathbb R)$? in $\{f\in C_0(\mathbb R), \ f(0)=0\}$?
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4$\begingroup$ Could you add a little context: where does this question come from, what have you tried or proved yet? $\endgroup$– cs89Commented Apr 18, 2023 at 12:21
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$\begingroup$ That seems to be a natural variant of the translates problem mathoverflow.net/questions/290631/… but i am not sure one can adpat the proof which is based on fourier analysis... $\endgroup$– davidCommented Apr 20, 2023 at 11:04
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