arXiv subjects
Publications and source records attributed to Dorin Ervin Dutkay.
We present a class of pairs of subsets of $\mathbb R^d$ for which the Fourier transform, when restricted to these subsets, is an isometric isomorphism, and thus the Plancherel identity is satisfied. The sets are invariant under translations by dual full-rank lattices.
In connection to the Fuglede conjecture, and to Fuglede's original work \cite{Fug74}, we study one-parameter unitary groups associated to self-adjoint extensions of the differential operator $Df=\frac1{2πi}f'$ on a union of finite intervals. We present a formula for such unitary groups and we use it to discover some geometric properties of such sets in $\br$ which admit orthogonal bases of exponential functions (also called spectral sets).
In connection with the Fuglede conjecture, we study the existence of commuting self-adjoint extensions of the partial differential operators on arbitrary, possibly disconnected domains in $\br^d$, the associated unitary group, the spectral measure and some geometric properties.
In relation to Fuglede's conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of $\mathbb{R}$ that are in duality. In the terminology commonly used in the context of Fuglede's conjecture, our result states that an open set tiles $\mathbb{R}$ by the finite set $\{0,1,\dots,p-1\}$ if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on $\left[-\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb{Z}$.
We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call {\it framenable}. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups $\mathbb{F}_n$, $\textup{Aut}(\mathbb{F}_2)$ and $\textup{Aut}(\mathbb{F}_3)$, all (countable) lattices of $SL(2,\mathbb{R})$, the Baumslag-Solitar groups $BS_{p,q}$, the braid groups $B_n$, and Thompson's group $F$.
The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of $L^2$ contexts where we make precise the interplay between the three sides of the question: (i) existence of orthogonal families of Fourier basis functions (and associated spectra) on the one hand, (ii) extensions of partial derivative operators, and (iii) geometry of the corresponding domains, stressing systems of translation-tiles. We emphasize an account of old and new developments since the original 1974-paper by Bent Fuglede where the co-authors and Steen Pedersen have contributed.
The purpose of the present paper is to place a number of geometric (and hands-on) configurations relating to spectrum and geometry inside a general framework for the {\it Fuglede conjecture}. Note that in its general form, the Fuglede conjecture concerns general Borel sets $Ω$ in a fixed number of dimensions $d$ such that $Ω$ has finite positive Lebesgue measure. The conjecture proposes a correspondence between two properties for $Ω$, one takes the form of spectrum, while the other refers to a translation-tiling property. We focus here on the case of dimension one, and the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the momentum operator $\frac{1}{2πi}\frac{d}{dx}$, realized in $L^2$ of a union of intervals.
A row co-isometry is a family $(V_i)_{i=0}^{N-1}$ of operators on a Hilbert space, subject to the relation $$\sum_{i=0}^{N-1}V_iV_i^*=I.$$ As shown in \cite{BJK00}, row co-isometries appear as compressions of representations of Cuntz algebras. In this paper we will present some general constructions of Parseval frames for Hilbert spaces, obtained by iterating the operators $V_i$ on a finite set of vectors. The constructions are based on random walks on finite graphs. As applications of our constructions we obtain Parseval Fourier bases on self-affine measures and Parseval Walsh bases on the interval. \end{abstract}
Motivated by the harmonic analysis of self-affine measures, we introduce a class of representations of the Cuntz algebra associated to random walks on graphs. The representations are constructed using the dilation theory of row coisometries. We study these representations, their commutant and the intertwining operators.
We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_ε$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, when the measure is the middle-third Cantor measure, or more generally for self-similar measures with no-overlap condition, there are always exponential Riesz sequences of maximal possible Beurling dimension.
We study Fourier bases on invariant measures generated by affine iterated function systems in ${\mathbb R}^d$ with integer coefficients. We show that, for simple digit sets, these systems satisfy the open set condition and have no overlap. We present natural geometric conditions under which such measures have an orthonormal basis or a frame of exponential functions with frequencies being a subset of ${\mathbb Z}^d$. Moreover, we characterize when such measures have a spectrum in ${\Bbb Z}^d$.
We present a way to construct Parseval frames of piecewise constant functions for $L^2[0,1]$. The construction is similar to the generalized Walsh bases. It is based on iteration of operators that satisfy a Cuntz-type relation, but without the isometry property. We also show how the Parseval frame can be dilated to an orthonormal basis and the operators can be dilated to true Cuntz isometries.
The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary group representations. The main purpose of this paper is present a more general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pairs. In particular, for the Gabor representations $π_Λ$ and $π_{Λ^{o}}$ with respect to a pair of dual time-frequency lattices $Λ$ and $Λ^{o}$ in $\R^{d}\times \R^{d}$ we have that $\{π_Λ(m, n)g_{1} \oplus ... \oplus π_Λ(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a frame for $L^{2}(\R^{d})\oplus ... \oplus L^{2}(\R^{d})$ if and only if $\cup_{i=1}^{k}\{π_{Λ^{o}}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a Riesz sequence, and $\cup_{i=1}^{k}\{π_Λ(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a frame for $L^{2}(\R^{d})$ if and only if $\{π_{Λ^{o}}(m, n)g_{1} \oplus ... \oplus π_{Λ^{o}}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a Riesz sequence. This appears to be new even in the context of Gabor analysis.
This paper continues the study of orthonormal bases (ONB) of $L^2[0,1]$ introduced in \cite{DPS14} by means of Cuntz algebra $\mathcal{O}_N$ representations on $L^2[0,1]$. For $N=2$, one obtains the classic Walsh system. We show that the ONB property holds precisely because the $\mathcal{O}_N$ representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal processing we find a fast generalized transform and compare this generalized transform with the classic one with respect to compression and sparse signal recovery.
Continuing the ideas from our previous paper, we construct Parseval frames of weighted exponential functions for self-affine measures.
We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number $m$ generates a complete or incomplete Fourier basis for a Cantor-type measure with scale $g$.
Based on tiles and on the Coven-Meyerowitz property, we present some examples and some general constructions of spectral subsets of integers.
We generalize an idea of Picioroaga and Weber to construct Paseval frames of weighted exponential functions for self-affine measures.