We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in $\{0, 1, 2, 3\}$) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.
A probability measure in R^d is called a spectral measure if it has an orthonormal basis consisting of exponentials. In this paper we study spectral Cantor measures. We establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral Cantor measure. This extends the earlier results of Jorgensen-Pedersen and Strichartz.
Let $μ_{q, b}$ be the Cantor measure associated with the iterated function system $f_i(x)=x/b+i/q, 0\le i\le q-1$, where $2\le q, b/q\in \Z$. In this paper, we consider spectra and maximal orthogonal sets of the Cantor measure $μ_{q, b}$ and their rational rescaling. We introduce a quantity to measure level difference between a branch and its subbranch for the labeling tree corresponding to a maximal orthogonal set of the Cantor measure $μ_{q, b}$, and use certain boundedness property of that quantity as sufficient and necessary conditions for a maximal orthogonal set of the Cantor measure $μ_…
In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical spectrum for the 1/4 Cantor measure always yields an alternate spectrum. We call this set an additive spectrum. The proof works by connecting the additive set to a spectrum formed by odd multiplicative scaling.
Nevanlinna--Pick norms on scattered and Cantor spectra
University of Warsaw Faculty of Mathematics, Informatics, and Mechanics（华沙大学数学、信息学和机械学院）；Institute of Mathematics Polish Academy of Sciences（波兰科学院数学研究所）
Cantor measures with odd base do not admit Fourier frames
Center for Data Science, New York University（纽约大学数据科学中心）；Department of Mathematics, Bar-Ilan University（巴伊兰大学数学系）；Department of Mathematics ETH Zürich（苏黎世联邦理工学院数学系）
In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} μ= μ(N_n,B_n) =& δ_{\frac{1}{N_1}B_1}\astδ_{\frac{1}{N_1N_2}B_2}\ast δ_{\frac{1}{N_1N_2N_3}B_3}\ast...\\ =& μ_n\astμ_{>n} \end{aligned} $$ with support inside $[0,1]$ always admits an exponential orthonormal basis $E(Λ) = \{e^{2πi λx}:λ\inΛ\}$ for $L^2(μ)$, where $Λ$ is obtained from suitably modifying $L_n$. Here, $μ_n$ is the convolution of the first $n$ Dirac measures …
The measure supported on the Cantor-4 set constructed by Jorgensen-Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. We construct Fourier frames for this measure via a dilation theory type construction. We expand the Cantor-4 set to a 2 dimensional fractal which admits a representation of a Cuntz algebra. Using the action of this algebra, an orthonormal set is generated on the larger fractal, which is then projected onto the Cantor-4 set to produce a Fourier frame.
We consider equally-weighted Cantor measures $μ_{q,b}$ arising from iterated function systems of the form ${b^{-1}(x+i)}$, $i=0,1,...,q-1$, where $q<b$. We classify the $(q,b)$ so that they have infinitely many mutually orthogonal exponentials in $L^2(μ_{q,b})$. In particular, if $q$ divides $b$, the measures have a complete orthogonal exponential system and hence spectral measures. We then characterize all the maximal orthogonal sets $Λ$ when $q$ divides $b$ via a maximal mapping on the $q-$adic tree in which all elements in $Λ$ are represented uniquely in finite $b-$adic expansions and we ca…
Each Cantor measure (μ) with scaling factor 1/(2n) has at least one associated orthonormal basis of exponential functions (ONB) for L^2(μ). In the particular case where the scaling constant for the Cantor measure is 1/4 and two specific ONBs are selected for L^2(μ), there is a unitary operator U defined by mapping one ONB to the other. This paper focuses on the case in which one ONB (Γ) is the original Jorgensen-Pedersen ONB for the Cantor measure (μ) and the other ONB is is 5Γ. The main theorem of the paper states that the corresponding operator U is ergodic in the sense that only the constan…
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.
The divergence of Mock Fourier series for spectral measures
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院计算与随机数学教育部重点实验室）
Based on the seminal work of Hutchinson, we investigate properties of {\em $α$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $α$ is a vector of nonnegative weights summing to $1$, and the corresponding weighted Cantor measure $μ^α$ is the unique Borel probability measure on $[0,1]$ satisfying $ μ^α(E) = \sum_{ n=0 }^{N-1} α_nμ^α( φ_n^{-1}(E) )$ where $φ_n: x\mapsto (x+n)/N$. In Sections 1 and 2 we examine several general properties of the measure $μ^α$ and the associated Legendre polynomials in $L_{μ^α}^2[0,1]$. In Section 3, we (1) compute the Lapl…
For $ξ\in \big( 0, \frac{1}{2} \big)$, let $E_ξ$ be the perfect symmetric set associated with $ξ$, that is $$E_ξ = \Big\{ \exp \Big( 2i π(1-ξ) \sum_{n = 1}^{+\infty} ε_{n} ξ^{n-1} \Big) : \, ε_{n} = 0 \textrm{ or } 1 \quad (n \geq 1) \Big\}$$ and $$b(ξ) = \frac{\log{\frac{1}ξ} - \log{2}}{2\log{\frac{1}ξ} - \log{2}}.$$ Let $q\geq 3$ be an integer and $s$ be a nonnegative real number. We show that any invertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$ that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big), \,n \rightarrow +\infty \\ & \textrm{a…
In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures.
We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the…
The paper is devoted to the development of general theory of packing measures and dimensions via introducing the notion of <<faithfulness of a packing family for $\dim_P$ calculation>> and the packing analogues of the Billingsley dimension. To this aim we study equivalent definitions of packing dimension and prove theorems which can be considered as packing analogues of the famous Billingsley's theorems. The main result of the paper gives necessary and sufficient condition for the packing dimension faithfulness of the family of cylinders generated by the Cantor series expansion. To the best of…
