Results 1  10
of
14
The study of translational tiling with Fourier Analysis
, 2003
"... Lectures given at the Workshop on Fourier Analysis and Convexity, Università Di Milano–bicocca ..."
Abstract

Cited by 24 (8 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
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.
Fuglede’s conjecture for a union of two intervals
, 2008
"... We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations. ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations.
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 ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
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.
ORTHOGONAL EXPONENTIALS, DIFFERENCE SETS, AND ARITHMETIC COMBINATORICS
"... Abstract. 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 #(Aq ≡ A ∩ [0, q]d). q. This extends and clarifies in the plane the result of Iosevich and Rudnev. As ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. 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 #(Aq ≡ A ∩ [0, q]d). q. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corrollary, we obtain the result from [IKP01] and [IKT01] that if K is a centrally symmetric convex body with a smooth boundary and nonvanishing curvature, then L2(K) does not possess an orthogonal basis of exponentials. We also give ground to the conjecture that the disk has not more than 3 orthogonal exponentials. This is done by proving that if a set A has a difference
ANALYSIS OF ORTHOGONALITY AND OF ORBITS IN AFFINE ITERATED FUNCTION SYSTEMS
, 2006
"... 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 ..."
Abstract
 Add to MetaCart
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 L 2 (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
unknown title
"... Abstract 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. ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract 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.
Notation. Let Ω ⊂ � d be measurable of measure 1. The Hilbert space L 2 (Ω)
"... Let Ω ⊆ � d be an open set of measure 1. An open set D ⊆ � d is called a ‘tight orthogonal packing region ’ for Ω if D − D does not intersect the zeros of the Fourier transform of the indicator function of Ω, and D has measure 1. Suppose that Λ is a discrete subset of � d. The main contribution of ..."
Abstract
 Add to MetaCart
(Show Context)
Let Ω ⊆ � d be an open set of measure 1. An open set D ⊆ � d is called a ‘tight orthogonal packing region ’ for Ω if D − D does not intersect the zeros of the Fourier transform of the indicator function of Ω, and D has measure 1. Suppose that Λ is a discrete subset of � d. The main contribution of this paper is a new way of proving the following result: D tiles � d when translated at the locations Λ if and only if the set of exponentials EΛ = {exp 2πi〈λ, x 〉 : λ ∈ Λ} is an orthonormal basis for L 2 (Ω). (This result has been proved by different methods by Lagarias, Reeds and Wang [9] and, in the case of Ω being the cube, by Iosevich and Pedersen [3]. When Ω is the unit cube in � d, it is a tight orthogonal packing region of itself.) In our approach, orthogonality of EΛ is viewed as a statement about ‘packing ’ � d with translates of a certain nonnegative function and, additionally, we have completeness of EΛ in L 2 (Ω) if and only if the abovementioned packing is in fact a tiling. We then formulate the tiling condition in Fourier analytic