arXiv subjects
Publications and source records attributed to Chun-Kit Lai.
We generalize the compatible tower condition given by Strichartz to the almost-Parseval-frame tower and show that non-trivial examples of almost-Parseval-frame tower exist. By doing so, we demonstrate the first singular fractal measure which has only finitely many mutually orthogonal exponentials (and hence it does not admit any exponential orthonormal bases), but it still admits Fourier frames.
By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.
A finite Borel measure $μ$ in ${\mathbb R}^d$ is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for $L^2(μ)$. It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures $ν$ and $λ$ without atoms whose supports form a packing pair, then $ν\ast λ+δ_t\astν$ is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure $μ_4+μ_{16}$ does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping $R_θ$ with $θ\ne \pmπ/2$, the two-dimensional measure $ρ_θ (\cdot): = (μ_4\timesδ_0)(\cdot)+(δ_0\timesμ_{16})(R_θ^{-1}\cdot)$, supported on the union of $x$-axis and $y=(\cot θ)x$, always admit a Fourier frame. Furthermore, we can find $\{e^{2πi \langleλ,x\rangle}\}_{λ\inΛ_θ}$ such that it forms a Fourier frame for $ρ_θ$ with frame bounds independent of $θ$. Nonetheless, $ρ_{\pmπ/2}$ does not admit any Fourier frame.
Let $R$ be an expanding matrix with integer entries and let $B,L$ be finite integer digit sets so that $(R,B,L)$ form a Hadamard triple on ${\br}^d$ in the sense that the matrix $$ \frac{1}{\sqrt{|\det R|}}\left[e^{2πi \langle R^{-1}b,\ell\rangle}\right]_{\ell\in L,b\in B} $$ is unitary. We prove that the associated fractal self-affine measure $μ= μ(R,B)$ obtained by an infinite convolution of atomic measures $$ μ(R,B) = δ_{R^{-1} B}\astδ_{R^{-2}B}\astδ_{R^{-3}B}\ast... $$ is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in $L^2(μ)$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.
Let $q\geq 2$ be an integer, and $\Bbb F_q^d$, $d\geq 1$, be the vector space over the cyclic space $\Bbb F_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E \subset \Bbb F_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2(\Bbb F_q^d)$. We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in $\Bbb F_q^d$, $d\geq 2$, $q$ an odd prime and $q\equiv 3$ (mod 4).
Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A, and then the unique sparsest solution exists, and recoverable by $\ell_1$-minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements Ax = b. Specifically, suppose A is of full spark with m rows, and suppose $\frac{m}{2}$ < s < m. Then the solution to Ax = b is unique for x with $\|x\|_0 \leq s$ up to a set of measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by an $\ell_1$-tail minimization procedure, which recovers sparse signals uniquely with s > $\frac{m}{2}$ in thousands and thousands of random tests. We further show instead that the mere $\ell_1$-minimization would actually fail if s > $\frac{m}{2}$ even from the same measure theoretical point of view.
If $x_1,\dots,x_m$ are finitely many points in $\mathbb{R}^d$, let $E_ε=\cup_{i=1}^m\,x_i+Q_ε$, where $Q_ε=\{x\in \mathbb{R}^d,\,\,|x_i|\le ε/2, \, i=1,...,d\}$ and let $\hat f$ denote the Fourier transform of $f$. Given a positive Borel measure $μ$ on $\mathbb{R}^d$, we provide a necessary and sufficient condition for the frame inequalities $$ A\,\|f\|^2_2\le \int_{\mathbb{R}^d}\,|\hat f(ξ)|^2\,dμ(ξ)\le B\,\|f\|^2_2,\quad f\in L^2(E_ε), $$ to hold for some $A,B>0$ and for some $ε>0$ sufficiently small. If $m=1$, we show that the limits of the optimal lower and upper frame bounds as $ε\rightarrow 0$ are equal, respectively, to the lower and upper Beurling density of $μ$. When $m>1$, we extend this result by defining a matrix version of Beurling density. Given a (possibly dense) subgroup $G$ of $\mathbb{R}$, we then consider the problem of characterizing those measures $μ$ for which the inequalities above hold whenever $x_1,\dots,x_m$ are finitely many points in $G$ (with $ε$ depending on those points, but not $A$ or $B$). We point out an interesting connection between this problem and the notion of well-distributed sequence when $G=a\,\mathbb{Z}$ for some $a>0$. Finally, we show the existence of a discrete set $Λ$ such that the measure $μ=\sum_λ\,δ_λ$ satisfy the property above for the whole group $\mathbb{R}$.
We study spectral measures generated by infinite convolution products of discrete measures generated by Hadamard triples, and we present sufficient conditions for the measures to be spectral, generalizing a criterion by Strichartz. We then study the spectral measures generated by random convolutions of finite atomic measures and rescaling, where the digits are chosen from a finite collection of digit sets. We show that in dimension one, or in higher dimensions under certain conditions, "almost all" such measures generate spectral measures, or, in the case of complete digit sets, translational tiles. Our proofs are based on the study of self-affine spectral measures and tiles generated by Hadamard triples in quasi-product form.
Let $R$ be an expanding matrix with integer entries and let $B,L$ be finite integer digit sets so that $(R,B,L)$ form a Hadamard triple on ${\br}^d$. We prove that the associated self-affine measure $μ= μ(R,B)$ is a spectral measure, which means it admits an orthonormal bases of exponential functions in $L^2(μ)$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors.
We consider the problem in determining the countable sets $Λ$ in the time-frequency plane such that the Gabor system generated by the time-frequency shifts of the window $χ_{[0,1]^d}$ associated with $Λ$ forms a Gabor orthonormal basis for $ L^2({\Bbb R}^d)$. We show that, if this is the case, the translates by elements $Λ$ of the unit cube in ${\Bbb R}^{2d}$ must tile the time-frequency space ${\Bbb R}^{2d}$. By studying the possible structure of such tiling sets, we completely classify all such admissible sets $Λ$ of time-frequency shifts when $d=1,2$. Moreover, an inductive procedure for constructing such sets $Λ$ in dimension $d\ge 3$ is also given. An interesting and surprising consequence of our results is the existence, for $d\geq 2$, of discrete sets $Λ$ with ${\mathcal G}(χ_{[0,1]^d},Λ)$ forming a Gabor orthonormal basis but with the associated "time"-translates of the window $χ_{[0,1]^d}$ having significant overlaps.
Let $Q$ be a fundamental domain of some full-rank lattice in ${\Bbb R}^d$ and let $μ$ and $ν$ be two positive Borel measures on ${\Bbb R}^d$ such that the convolution $μ\astν$ is a multiple of $χ_Q$. We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated $L^2$ space admits an orthogonal basis of exponentials) and we show that this is the case when $Q = [0,1]^d$. This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede's conjecture for spectral measures on ${\Bbb R}^1$ and we show that it implies the classical Fuglede's conjecture on ${\Bbb R}^1$.
Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^αq$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^α$ and $pq$.
Given discrete subsets $Λ_j\subset {\Bbb R}^d$, $j=1,...,q$, consider the set of windowed exponentials $\bigcup_{j=1}^{q}\{g_j(x)e^{2πi <λ,x>}: λ\inΛ_j\}$ on $L^2(Ω)$. We show that a necessary and sufficient condition for the windows $g_j$ to form a frame of windowed exponentials for $L^2(Ω)$ with some $Λ_j$ is that $m\leq \max_{j\in J}|g_j|\leq M$ almost everywhere on $Ω$ for some subset $J$ of $\{1,..., q\}$. If $Ω$ is unbounded, we show that there is no frame of windowed exponentials if the Lebesgue measure of $Ω$ is infinite. If $Ω$ is unbounded but of finite measure, we give a sufficient condition for the existence of Fourier frames on $L^2(Ω)$. At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.
The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on ${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven-Meyerowitz property for finite sets of integers, introduced in \cite{CoMe99}, and we show that if the spectral sets and the tiles in ${\mathbb Z}$ satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on ${\mathbb R}^1$ are true.
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 can separate the maximal orthogonal sets into two types: regular and irregular sets. For a regular maximal orthogonal set, we show that its completeness in $L^2(μ_{q,b})$ is crucially determined by the certain growth rate of non-zero digits in the tail of the $b-$adic expansions of the elements. Furthermore, we exhibit complete orthogonal exponentials with zero Beurling dimensions. These examples show that the technical condition in Theorem 3.5 of \cite{[DHSW]} cannot be removed. For an irregular maximal orthogonal set, we show that under some condition, its completeness is equivalent to that of the corresponding regularized mapping.
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 ${\mathcal G}(g,Λ,{\mathcal J})$.
We study the structure of the digit sets ${\mathcal D}$ for the integral self-similar tiles $T(b,{\mathcal{D}})$ (we call such ${\mathcal D}$ a {\it tile digit set} with respect to $b$). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial $P_{\mathcal D}$, i.e., the zeros of $P_{\mathcal D}$ on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomial, we characterize the tile digit sets through some product of cyclotomic polynomials (kernel polynomials), which is a generalization of the product-form to higher order.
Let $μ$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(μ)$. We show that if $L^2(μ)$ admits an exponential frame, then $μ$ must be of pure type. We also classify various $μ$ that admits either kind of exponential bases, in particular, the discrete measures and their connection with integer tiles. By using this and convolution, we construct a class of singularly continuous measures that has an exponential Riesz basis but no exponential orthonormal basis. It is the first of such kind of examples.