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Dec 20, 2013 at 20:03 comment added Peter Luthy 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.
Dec 20, 2013 at 18:35 history edited mkreisel CC BY-SA 3.0
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Dec 20, 2013 at 18:28 history edited mkreisel CC BY-SA 3.0
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Dec 20, 2013 at 18:27 comment added mkreisel I'll amend the question to specify that $K$ should be compact.
Dec 20, 2013 at 17:00 comment added Alexandre Eremenko Nothing like this can be true, even when $d=1$, if the set $K$ is unbounded.
Dec 20, 2013 at 16:58 answer added user6891 timeline score: 2
Dec 20, 2013 at 15:06 answer added Dirk timeline score: 2
Dec 20, 2013 at 14:45 history asked mkreisel CC BY-SA 3.0