In this paper, we study the class of tempered distributions whose Fourier transform is a translation bounded measure and show that each such distribution in Rd${\mathbb {R}}^d$ has order at most 2d. ...We show the existence of the generalized Eberlein decomposition within this class of distributions, and its compatibility with all previous Eberlein decompositions. The generalized Eberlein decomposition for Fourier transformable measures and properties of its components are discussed. Lastly, we take a closer look at the absolutely continuous spectrum of measures supported on Meyer sets.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
In this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes ...continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of the Fourier transform on the cone of positive definite measures. In the appendix, we characterize vague precompactness of a set of measures in arbitrary LCAG, and the necessity of second countability property of a group for defining the autocorrelation measure.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we prove that given a cut-and-project scheme
$(G, H, \mathcal {L})$
and a compact window
$W \subseteq H$
, the natural projection gives a bijection between the Fourier transformable ...measures on
$G \times H$
supported inside the strip
${\mathcal L} \cap (G \times W)$
and the Fourier transformable measures on G supported inside
${\LARGE \curlywedge }(W)$
. We provide a closed formula relating the Fourier transform of the original measure and the Fourier transform of the projection. We show that this formula can be used to re-derive some known results about Fourier analysis of measures with weak Meyer set support.
A separated sequence
Λ on the real line is called a Pólya sequence if any entire function of zero exponential type bounded on
Λ is constant. In this paper we solve the problem by Pólya and Levinson ...that asks for a description of Pólya sets. We also show that the Pólya–Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and De Branges spaces of entire functions.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Any positive semi-definite function defined on Z (resp. R) can be represented as the Fourier transform of a positive Radon measure on T (resp. R). We give a proof of this celebrated result due to ...Herglotz and Bochner from the viewpoint of Schwartz's theory of distributions.
In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian ...groups.
The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in R2d, d≥3. These surfaces are defined by a complex curve γ(z) of simple type, which is given ...by a mapping of the form z↦γ(z)=(z,z2,…,zd−1,ϕ(z)) where ϕ(z) is an analytic function on a domain Ω⊂C. This is regarded as a real mapping z=(x,y)↦γ(x,y) from Ω⊂R2 to R2d.
Our results cover the case ϕ(z)=zN for any nonnegative integer N, in all dimensions d≥3. The main result is a uniform estimate, valid when d=3, where ϕ(z) may be taken to be an arbitrary polynomial of degree at most N. It is uniform in the sense that the operator norm is independent of the coefficients of the polynomial. These results are analogues of the uniform restricted strong type estimates in 5, valid for polynomial curves of simple type and some other classes of curves in Rd, d≥3.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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