Some properties of spectral measures
type: Item
instance of: scholarly article
title: Some properties of spectral measures
zbMATH Open document ID: 1096.42017
author: Izabella Łaba
author: Yang Wang
published in: Applied and Computational Harmonic Analysis
publication date: 2006-03-20T00:00:00Z
review text: Spectral measures, first introduced by \textit{P. E. T. Jørgensen} and \textit{S. Pedersen} [J.\ Anal.\ Math.\ 75, 185--228 (1998; Zbl 0959.28008)], are a natural extension of spectral sets. A finite Borel measure \(\mu\) in \(R^d\) is called a \textit{spectral measure} if there exists a set \(\Lambda \subset R^d\) such that the set of exponentials \(\{\exp(2 \pi i \lambda \cdot x); \lambda \in \Lambda\}\) forms an orthogonal basis for \(L^2(\mu)\). Let \(\Omega\) be a measurable set in \(R^d\), and let \(\mu\) denote the restriction of the \(d\)--dimensional Lebesgue measure to \(\Omega\). We say that \(\Omega\) is a \textit{spectral set} if \(\mu\) is a spectral measure. The main interest for studying spectral sets comes from their misterious connection to tiling, first formulated in a conjecture by \textit{B. Fuglede} [J. Funct. Anal. 16, 101--121 (1974; Zbl 0279.47014)], and known today as the Fudledge Conjecture or the Spectral Set Conjecture: A measurable set \(\Omega\) in \(R^d\) is a spectral set if and only if it tiles \(R^d\) by translation. The conjecture had baffled the mathematicians that study spectral sets for years until very recently, when \textit{T. Tao} [Math. Res. Lett. 11, No. 2--3, 251--258 (2004; Zbl 1092.42014)] exhibited a spectral set in dimensions \(d \geq 5\) that is not a tile, and \textit{M. N. Kolountzakis} and \textit{M. Matolcsi} [``Tiles with no spectra'', preprint, arXiv: math.CA/0406127] exhibited tiles that are not spectral sets in dimensions \(d \geq 5\). Despite the counterexamples, the connection between spectral sets and tiling is strongly evident, especially in low dimensions, as indicated by earlier works, including the original article by Fuglede. The study of spectral measures so far has focused on self--similar measures. These measures have the advantage that their Fourier transforms can be expressed as infinite products, which permits the computation of their zeros. In the article under review the authors study measures…
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5013516
zbMATH Keywords: spectral measure
zbMATH Keywords: Fuglede conjecture
zbMATH Keywords: spectral pair
zbMATH Keywords: tiling
reviewed by: Richard A. Zalik
MaRDI profile type: Publication
full work available at URL: j.acha.2005.03.003
OpenAlex ID: W2042068720
cites work: Q5833361
cites work: Commuting self-adjoint partial differential operators and a group theoretic problem
cites work: The Fuglede spectral conjecture holds for convex planar domains
cites work: Dense analytic subspaces in fractal \(L^2\)-spaces
cites work: Tiles with no spectra
cites work: Fuglede's conjecture for a union of two intervals
cites work: On spectral Cantor measures
cites work: Spectral sets and factorizations of finite abelian groups
cites work: Necessary density conditions for sampling an interpolation of certain entire functions
cites work: Universal spectra, universal tiling sets and the spectral set conjecture
cites work: Mock Fourier series and transforms associated with certain Cantor measures
cites work: Fuglede's conjecture is false in 5 and higher dimensions
DOI: 10.1016/J.ACHA.2005.03.003
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description: scientific article; zbMATH DE number 5013516