The Remez inequality for linear combinations of shifted Gaussians
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abstract
Let |f|:= supt|f(t)| and Gn:= f: f(t) = j=1n aje-(t-j)2, aj, j }. We prove that there is an absolute constant c1 > 0 such that [ exp (c1 (min{n{1/2}s, ns2} + s 2) f exp (80(min{n{1/2}s, ns2} + s2) for every s (0, ) and n 9, where the supremum is taken for all f Gn with m ({t }|f(t)| 1}) 7le; s. This is what we call (an essentially sharp) Remez-type inequality for the class Gn. We also prove the right higher dimensional analog of the above result. Cambridge Philosophical Society 2008.