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The study of translational tiling with Fourier Analysis
, 2003
"... Lectures given at the Workshop on Fourier Analysis and Convexity, Università Di Milano–bicocca ..."
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Cited by 15 (4 self)
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Lectures given at the Workshop on Fourier Analysis and Convexity, Università Di Milano–bicocca
FOURIER ANALYSIS AND GEOMETRIC COMBINATORICS
, 2004
"... This article is based on the series of lectures on the interaction of Fourier analysis and geometric combinatorics delivered by the author in Padova at the Minicorsi di Analisi Matematica in June, 2002. ..."
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Cited by 7 (3 self)
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This article is based on the series of lectures on the interaction of Fourier analysis and geometric combinatorics delivered by the author in Padova at the Minicorsi di Analisi Matematica in June, 2002.
ANALYSIS OF ORTHOGONALITY AND OF ORBITS IN AFFINE ITERATED FUNCTION SYSTEMS
"... Abstract. We introduce a duality for Affine Iterated Function Systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by sc ..."
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Abstract. We introduce a duality for Affine Iterated Function Systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure µ, and we ask when µ is a spectral measure, i.e., when the Hilbert space L2 (µ) has an orthonormal basis (ONB) of exponentials {eλ  λ ∈ Λ}. We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in
DISTANCES SETS THAT ARE A SHIFT OF THE INTEGERS AND FOURIER BASIS FOR PLANAR CONVEX SETS
, 709
"... Abstract. The aim of this paper is to prove that if a planar set A has a difference set ∆(A) satisfying ∆(A) ⊂ Z + + s for suitable s than A has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if A is a ..."
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Abstract. The aim of this paper is to prove that if a planar set A has a difference set ∆(A) satisfying ∆(A) ⊂ Z + + s for suitable s than A has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere nonvanishing curvature, then #(A ∩ [−q, q] 2) ≤ C(K)q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [IKP01] and [IKT01] that if K is a centrally symmetric convex body with a smooth boundary and
DUALITY QUESTIONS FOR OPERATORS, SPECTRUM AND MEASURES
, 2008
"... We explore spectral duality in the context of measures in R n, starting with partial differential operators and Fuglede’s question (1974) about the relationship between orthogonal bases of complex exponentials in L²(Ω) and tiling properties of Ω, then continuing with affine iterated function system ..."
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We explore spectral duality in the context of measures in R n, starting with partial differential operators and Fuglede’s question (1974) about the relationship between orthogonal bases of complex exponentials in L²(Ω) and tiling properties of Ω, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in R n, formulated first by Jorgensen and Pedersen.