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Dorin Ervin Dutkay

Publications and source records attributed to Dorin Ervin Dutkay.

At least 55 records · Page 3Linked to original sources
arXiv2012

Continuous and Discrete Fourier Frames for Fractal Measures

Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.

math.FA↗
arXiv2012

Uniformity of measures with Fourier frames

We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. To be more precise, by considering certain absolute continuity properties of the measure and its translation, we recover the characterization on absolutely continuous measures $g\, dx$ with Fourier frames obtained in \cite{Lai11}. Moreover, we prove that the frame bounds are pushed away by the essential infimum and supremum of the function $g$. This also shows that absolutely continuous spectral measures supported on a set $Ω$, if they exist, must be the standard Lebesgue measure on $Ω$ up to a multiplicative constant. We then investigate affine iterated function systems (IFSs), we show that if an IFS with no overlap admits a frame measure then the probability weights are all equal. Moreover, we also show that the Łaba-Wang conjecture \cite{MR1929508} is true if the self-similar measure is absolutely continuous. Finally, we will present a new approach to the conjecture of Liu and Wang \cite{LW} about the structure of non-uniform Gabor orthonormal bases of the form ${\mathcal G}(g,Λ,{\mathcal J})$.

math.FA↗
arXiv2012

Iterative approximations of exponential bases on fractal measures

For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra. In this paper we investigate this problem for general fractal measures induced by iterated function systems (IFS). We prove some existence results of spectra associated with Hadamard pairs. We also obtain some characterizations of Bessel spectrum in terms of finite matrices for affine IFS measures, and one sufficient condition of frame spectrum in the case that the affine IFS has no overlap.

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arXiv2011

Decomposition of wavelet representations and Martin boundaries

We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms. Our main results offer a direct integral decomposition for the general wavelet representation, and we solve a question posed by Judith Packer. This entails a direct integral decomposition of the general wavelet representation. We further give a detailed analysis of the measures contributing to the decomposition into irreducible representations. We prove results for associated Martin boundaries, relevant for the understanding of wavelet filters and induced random-walks, as well as classes of harmonic functions. Our setting entails representations built from certain finite-to-one endomorphisms $r$ in compact metric spaces $X$, and we study their dilations to automorphisms in induced solenoids. Our wavelet representations are covariant systems formed from the dilated automorphisms. They depend on assigned measures $μ$ on $X$. It is known that when the data $(X, r, μ)$ are given the associated wavelet representation is typically reducible. By introducing wavelet filters associated to $(X, r)$ we build random walks in $X$, path-space measures, harmonic functions, and an associated Martin boundary. We construct measures on the solenoid $(X_\infty, r_\infty)$, built from $(X, r)$. We show that $r_\infty$ induces unitary operators $U$ on Hilbert space $\H$ and representations $π$ of the algebra $L^\infty(X)$ such that the pair $(U, r_\infty)$, together with the corresponding representation $π$ forms a cross-product in the sense of $C^*$-algebras. We note that the traditional wavelet representations fall within this wider framework of $(\H, U, π)$ covariant crossed products.

math.FA↗
arXiv2011

Fourier duality for fractal measures with affine scales

For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $\br^d$, and they both have the same matrix scaling. But the two use different translation vectors, one by a subset $B$ in $\br^d$, and the other by a related subset $L$. Among other things, we show that there is then a pair of infinite discrete sets $Γ(L)$ and $Γ(B)$ in $\br^d$ such that the $Γ(L)$-Fourier exponentials are orthogonal in $L^2(μ_B)$, and the $Γ(B)$-Fourier exponentials are orthogonal in $L^2(μ_L)$. These sets of orthogonal "frequencies" are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line. Our duality pairs do not always yield orthonormal Fourier bases in the respective $L^2(μ)$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.

math.FA↗
arXiv2011

Isospectral measures

In recent papers a number of authors have considered Borel probability measures $μ$ in $\br^d$ such that the Hilbert space $L^2(μ)$ has a Fourier basis (orthogonal) of complex exponentials. If $μ$ satisfies this property, the set of frequencies in this set are called a spectrum for $μ$. Here we fix a spectrum, say $Γ$, and we study the possibilities for measures $μ$ having $Γ$ as spectrum.

math.FA↗
arXiv2010

Irreducible wavelet representations and ergodic automorphisms on solenoids

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in \cite{DuJo06b}, and we present several equivalent formulations of the problem.

math.FA↗
arXiv2010

Bessel sequences of exponentials on fractal measures

Jorgensen and Pedersen have proven that a certain fractal measure $ν$ has no infinite set of complex exponentials which form an orthonormal set in $L^2(ν)$. We prove that any fractal measure $μ$ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in $L^2(μ)$ such that the frequencies have positive Beurling dimension.

math.FA↗
arXiv2010

Affine fractals as boundaries and their harmonic analysis

We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these fractals. We prove that as sets these fractals arise as boundaries of functions in closed subspaces of the Hardy space $H^2$. By this we mean that there are lacunary subsets $Γ$ of the non-negative integers, and associated closed $Γ$-subspace in the Hardy space $H^2(\bd)$, $\bd$ denoting the disk, such that for every function $f$ in in $H^2(Γ)$, and for every point $z$ in $\bd$, $f(z)$ admits a boundary integral represented by an associated measure $μ$, with integration over $\suppμ$ placed as a Cantor subset on the circle $\bt := \{bd}(\bd)$. We study families of pairs: measures $μ$ and sets $Γ$ of lacunary form, admitting lacunary Fourier series in $L^2(μ)$; i.e., configurations $Γ$ arranged with a geometric progression of empty spacing, or missing parts, gaps. Given $Γ$, we find corresponding generalized Szeg\" o kernels $G_Γ$, and we compare them to the classical Szeg\" o kernel for $\bd$. Rather than the more traditional approach of starting with $μ$, and then asking for possibilities for sets $Γ$, such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset $Γ$, and, within a new duality framework, we study the possibilities for choices of measures $μ$.

math.FA↗
arXiv2010

On the Beurling dimension of exponential frames

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff dimension of the fractal.

math.FA↗
arXiv2010

Spectral measures and Cuntz algebras

We consider a family of measures $μ$ supported in $\br^d$ and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in $L^2(μ)$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $Γ$ in $\br^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $Γ$ (spectrum) and the measures $μ$ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $C^*$-algebras $\mathcal O_N$.

math.FA↗
arXiv2009

Orthogonal exponentials, translations, and Bohr completions

We are concerned with an harmonic analysis in Hilbert spaces $L^2(μ)$, where $μ$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_λ(x) = \exp(2πi λx)$ in $L^2(μ)$. This question in turn is connected to the existence of a natural embedding of $L^2(μ)$ into an $L^2$-space of Bohr almost periodic functions on $\br^n$. In particular we explore when $L^2(μ)$ contains an orthogonal basis of $e_λ$ functions, for $λ$ in a suitable discrete subset in $\br^n$; i.e, when the measure $μ$ is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures $μ$ that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in $\br^n$. We show in this case that if $μ$ is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in $μ$. This solves part of the Łaba-Wang conjecture. As an application of the new non-overlap result, we solve the spectral-pair problem for Bernoulli convolutions advancing in this way a theorem of Ka-Sing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification.

math.FA↗
arXiv2009

A Duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for $L^{2}(\R^{d})$ if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for $\rm II_{1}$ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when $G$ is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well known problem about the classification of free group von Neumann algebras.

math.FA↗
arXiv2008

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of norm $\frac{1}{\sqrt{N}}$, the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by an Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left regular representation of a free group has this property.

math.FA↗
arXiv2008

Probability and Fourier duality for affine iterated function systems

Let $d$ be a positive integer, and let $μ$ be a finite measure on $\br^d$. In this paper we ask when it is possible to find a subset $Λ$ in $\br^d$ such that the corresponding complex exponential functions $e_λ$ indexed by $Λ$ are orthogonal and total in $L^2(μ)$. If this happens, we say that $(μ, Λ)$ is a spectral pair. This is a Fourier duality, and the $x$-variable for the $L^2(μ)$-functions is one side in the duality, while the points in $Λ$ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures $μ$ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in $\br^d$; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.

math.FA↗
arXiv2008

Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids

For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over $\bz$. Our starting point is a given pair $(A, \mathcal D)$ with the matrix $A$ assumed expansive, and $\mathcal D$ a chosen complete digit set, i.e., in bijective correspondence with the points in $\bz^d/A^T\bz^d$. We give an explicit geometric representation and encoding with infinite words in letters from $\mathcal D$. We show that the attractor $X(A^T,\mathcal D)$ for an affine Iterated Function System (IFS) based on $(A,\mathcal D)$ is a set of fractions for our digital representation of points in $\br^d$. Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid $\sa$ associated with the pair $(A,\mathcal D)$. The intricate part (Theorem \ref{thenccycl}) is played by the cycles in $\bz^d$ for the initial $(A,\mathcal D)$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\mathcal D$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces.

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