Suppose we have a relatively dense collection of points $\Lambda \subset \mathbb{R}^d$ and $K \subset \mathbb{R}^d$ where $K$ is compact and measurable. When will the linear span of the collection of exponentials $\{e^{2\pi i \lambda \cdot x}\}_{\lambda \in \Lambda}$ be dense in $L^2(K)?$ There should be some necessary condition like $|K|<\overline{dens}(\Lambda)$ where $\overline{dens}(\Lambda)$ is the upper density of $\Lambda$ (please let me know if I've got this backwards and it should be the lower density). Are there any easy sufficient conditions?
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$\begingroup$ Nothing like this can be true, even when $d=1$, if the set $K$ is unbounded. $\endgroup$– Alexandre EremenkoDec 20, 2013 at 17:00
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$\begingroup$ I'll amend the question to specify that $K$ should be compact. $\endgroup$– mkreiselDec 20, 2013 at 18:27
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1$\begingroup$ Alexei Poltoratski gave a very nice series of lectures with a number of comments in this direction this summer at Clemson. I don't think he really discussed the R^d case, but notes similar to the lectures he gave are available here: internetanalysisseminar.gatech.edu/lectures_uncp Some of the references may point you in the right direction. $\endgroup$– Peter LuthyDec 20, 2013 at 20:03
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2 Answers
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Not a full answer, but a question like this stated and answered for $K=]-\infty,\infty[$ in "A class of nonharmonic Fourier series", R. J. Duffin and A. C. Schaeffer, Trans. Amer. Math. Soc. 72 (1952), 341-366 initiated the theory of frames. Your question is answered for $K = [-A,A]$ in "An Introduction to Nonharmonic Fourier Series" by Robert M. Young.
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$\begingroup$ Thanks! I am especially interested in the question for $K \subset \mathbb{R}^d, d>1.$ $\endgroup$– mkreiselDec 20, 2013 at 15:23
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Just a comment but I am not empowered. Firstly don't you mean something like the linear hull being dense? In general, the type of family you describe will not be dense. If we use this recasting of your query and $K$ is bounded, then the Fourier transform of any function in your space will be holomorphic and standard duality arguments show that it would suffice that your set be one of uniqueness for such functions. The same argument can be used to get more subtle results by using the Payley-Wiener theorem and standard facts on the zeroes of entire functions which satisfy suitable growth conditions (being of exponential type).
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$\begingroup$ Yes, I meant to ask when the linear span of the set $\{e^{2 \pi i \lambda}\}$ would be dense. Certainly the Paley-Wiener theorem relates $L^2(K)$ to holomorphic functions with certain growth conditions. What do you mean by a "set of uniqueness?" $\endgroup$– mkreiselDec 20, 2013 at 18:35
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$\begingroup$ A set of uniqueness for a family of (say holomorphic) functions is a subset of their domain which is such that if a function from this family vanishes on this set, then it is identically zero. $\endgroup$– user6891Dec 20, 2013 at 19:18
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$\begingroup$ Ah thanks. Are there particular techniques for proving that a set is a set of uniqueness? $\endgroup$– mkreiselDec 20, 2013 at 19:21
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1$\begingroup$ There are whole books on this. I would suggest "Entire Functions" by Ralph Boas for starters. $\endgroup$– user6891Dec 20, 2013 at 19:43
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$\begingroup$ This book seems concentrated on the $d=1$ case. Has a similar analysis been done in higher dimensions? $\endgroup$– mkreiselDec 20, 2013 at 20:52