Download e-book for iPad: Applications to Fourier series (2005)(en)(4s) by Garrett P.

Download e-book for iPad: Applications to Fourier series (2005)(en)(4s) by Garrett P.

By Garrett P.

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Extra info for Applications to Fourier series (2005)(en)(4s)

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The Hruscev theorem stated above is just a representative of a long series of his results on "sparse supports & unilateral decay" including many concrete sorts of "decay" and various "objects" (not necessarily charges); only a part of the results of [Hm] is in [HJ]. 3. Local and Antilocal Convolutions This part is devoted to a form of the UP for shift invariant linear operators; we call it antilocality. 3. 3. In this part everything is closely related to the UP of Parts 1 and 2. 1. Local and completely local convolutions We denote by V' (JRd) the set of all distributions in JRd.

Four theorems on the unilateral decay This series of theorems starts historically with the following result due to Levinson and Cartwright: (I) Suppose fEU (1l') satisfies lj(n)1 :::; h(lnl) for all negative integers n, (21) h : [1, +(0) --t (0, +(0) being a decreasing function ("unilateral decay of j"). If ~ log h(n) __ (22) ~ n2 - 00, n=l then f cannot vanish identically on a non-degenerate arc unless f = O. p. Havin / On the Uncertainty Principle in Harmonic Analysis then jl cannot vanish identically on a set of positive length unless I" = O.

It is intuitively plausible that the mUltiplicity of this block shift is one, but that is no tautology, and indeed verification of this is the heart of the proof. We must show that the wandering space W is one-dimensional. To this end, let cp denote any vector in W, which in our context is identified as M e ei8 M. Thus, cp is in M, and orthogonal to ei8 f to every f in M. We can in particular choose f = eik8 cp for k = 0,1,2, ... and this gives vn ! Icpl2 ein8 dO = 0 n = 1,2, .... 11' By complex conjugation this holds also for all negative integers n, hence almost everywhere on 'll'.

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