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Moduli of continuity and average decay of Fourier transforms: twosided estimates, submitted. math.CA/0212254
"... Abstract. We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1 below). We note that our approach does not use a regularly varying com ..."
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Cited by 4 (2 self)
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Abstract. We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1 below). We note that our approach does not use a regularly varying comparison function as in [2]. A corollary of Theorem 1.1 deals with the equivalence of the twosided estimates on the modulus of continuity on one hand, and on the tail of the Fourier transform, on the other (Corollary 1.5). This corollary is applied in the proof of the violation of the socalled entropic area law for a critical system of free fermions in [4, 5].
A SPECIAL CASE OF A CONJECTURE BY WIDOM WITH IMPLICATIONS TO FERMIONIC ENTANGLEMENT ENTROPY
, 906
"... Abstract. We prove a special case of a conjecture by Harold Widom. More precisely, we establish the leading and nexttoleading term of a semiclassical expansion of the trace of the square of certain integral operators on the Hilbert space L 2 (R d). As already observed by Gioev and Klich, this imp ..."
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Cited by 4 (1 self)
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Abstract. We prove a special case of a conjecture by Harold Widom. More precisely, we establish the leading and nexttoleading term of a semiclassical expansion of the trace of the square of certain integral operators on the Hilbert space L 2 (R d). As already observed by Gioev and Klich, this implies a logarithmically enhanced “area law ” of the entanglemententropy of the free Fermi gas in its ground state for large scales. 1.
On Hankeltype operators with discontinuous symbols in higher dimensions
 Bull. London Math. Soc. 44(2012), Issue
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Edge Illumination and Imaging of Extended Reflectors
 SIAM J. IMAGING SCIENCES, VOL. 1, NO. 1, PP. 75–114
, 2008
"... We use the singular value decomposition of the array response matrix, frequency by frequency, to image selectively the edges of extended reflectors in a homogeneous medium. We show with numerical simulations in an ultrasound regime, and analytically in the Fraunhofer diffraction regime, that informa ..."
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We use the singular value decomposition of the array response matrix, frequency by frequency, to image selectively the edges of extended reflectors in a homogeneous medium. We show with numerical simulations in an ultrasound regime, and analytically in the Fraunhofer diffraction regime, that information about the edges is contained in the singular vectors for singular values that are intermediate between the large ones and zero. These transition singular vectors beamform selectively from the array onto the edges of the reflector crosssection facing the array, so that these edges are enhanced in imaging with traveltime migration. Moreover, the illumination with the transition singular vectors is done from the sources at the edges of the array. The theoretical analysis in the Fraunhofer regime shows that the singular values transition to zero at the index N ⋆ (ω) =AB/(λL) 2. Here A  and B  are the areas of the array and the reflector crosssection, respectively, ω is the frequency, λ is the wavelength, and L is the range. Since (λL) 2 /A  is the area of the focal spot size at range L, we see that N ⋆ (ω) is the number of focal spots contained in the reflector crosssection. The ultrasound simulations are in an extended Fraunhofer regime. The simulation results are, however, qualitatively similar to those obtained theoretically in the Fraunhofer regime. The numerical simulations indicate, in addition, that the subspaces spanned by the transition singular vectors are robust with respect to additive noise when the array has a large number of elements.
doi:10.1093/imrn/rnq085 A Special Case of a Conjecture by Widom with Implications to
, 2010
"... We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and nexttoleading term of a semiclassical expansion of the trace of the square of certain integral operators on the Hilbert space L2(Rd). As already observed by Gioev and Klich, ..."
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We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and nexttoleading term of a semiclassical expansion of the trace of the square of certain integral operators on the Hilbert space L2(Rd). As already observed by Gioev and Klich, this implies that the bipartite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement. 1