Analysis of \(\mu_{R,D}\)-orthogonality in affine iterated function systems
type: Item
instance of: scholarly article
title: Analysis of \(\mu_{R,D}\)-orthogonality in affine iterated function systems
zbMATH Open document ID: 1172.28005
author: Yan-Bo Yuan
published in: Acta Applicandae Mathematicae
publication date: 2009-04-29T00:00:00Z
review text: Let \(R\in M_n(\mathbb{Z})\) be an expanding integer matrix, \(D\subseteq\mathbb{Z}^n\) a finite subset and \(\mu_{R,D}\) a corresponding self-affine measure. The first main result in this paper, Theorem 2.3, concerns \(\mu_{R,D}\)-orthogonality and compatible pair conditions. This is some refined version of an earlier result [\textit{J.-L. Li}, J. Funct. Anal. 244, No. 2, 628--638 (2007; Zbl 1119.42014)] in a special case, but with the essential advantage to omit some seemingly artificial condition (already conjectured before). In Theorem~2.5 another conjecture, cf. [\textit{I. Łaba} and \textit{Y. Wang}, J. Funct. Anal. 193, No. 2, 409--420 (2002; Zbl 1016.28009)] about the equality of probability weights in case of a self-affine measure associated with an IFS can be verified in case of \(|D|=2\). Finally, in Section~3 the generalized plane Sierpiński gasket corresponding to \[ R=\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}\qquad\text{and}\qquad D=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right\} \] is studied. In Theorem~3.3 the author shows that then for \(\mu_{R,D}\) there are at most 3 mutually orthogonal exponential functions in \(L^2(\mu_{R,D})\) and the number 3 is best.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5549551
zbMATH Keywords: compatible pair
zbMATH Keywords: iterated function system
zbMATH Keywords: spectrum
zbMATH Keywords: spectral measure
zbMATH Keywords: Sierpinski gasket
reviewed by: Dorothee D. Haroske
MaRDI profile type: Publication
full work available at URL: s10440-008-9247-x
OpenAlex ID: W2025056579
cites work: Iterated function systems, Ruelle operators, and invariant projective measures
cites work: Fourier frequencies in affine iterated function systems
cites work: Analysis of orthogonality and of orbits in affine iterated function systems
cites work: Commuting self-adjoint partial differential operators and a group theoretic problem
cites work: Orthogonal exponentials on the ball
cites work: Self-similar lattice tilings
cites work: Q3731958
cites work: Distance sets that are a shift of the integers and Fourier basis for planar convex sets
cites work: Q4435799
cites work: Fourier bases and a distance problem of Erdős
cites work: Convex bodies with a point of curvature do not have Fourier bases
cites work: Dense analytic subspaces in fractal \(L^2\)-spaces
cites work: Orthogonal Exponentials for Bernoulli Iterated Function Systems
cites work: Non-symmetric convex domains have no basis of exponentials
cites work: On spectral Cantor measures
cites work: Some properties of spectral measures
cites work: On characterizations of spectra and tilings
cites work: \(\mu _{m,d}\)-orthogonality and compatible pair
cites work: SPECTRAL SELF-AFFINE MEASURES IN $\mathbb{R}^N$
cites work: Remarks on ``Dense analytic subspaces in fractal \(L^2\)-spaces'' by P. E. T. Jorgensen and S. Pedersen
cites work: Mock Fourier series and transforms associated with certain Cantor measures
DOI: 10.1007/S10440-008-9247-X
description: scientific article; zbMATH DE number 5549551