Gady Kozma, Shahaf Nitzan, Alexander Olevskii
We show that there exists a bounded subset of R such that no system of exponentials can be a Riesz basis for the corresponding Hilbert space. An additional result gives a lower bound for the Riesz constant of any putative Riesz basis of the two dimensional disk.
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Unions of exponential Riesz bases
本文提出通过频率集和定义域的并集构造指数Riesz基的新方法，并给出示例说明其易用性及指数Riesz基的微妙性质。
Exponential Riesz bases in non-Archimedean locally compact Abelian groups
本文研究了非阿基米德局部紧阿贝尔群中指数Riesz基的存在性，证明了所有球的有限并集存在该基，而某些有界集不存在此类基。
On the lower Riesz basis bound of exponential systems over an interval
本文重新审视Pavlov的指数Riesz基刻画，研究下Riesz基界，并改进Avdonin定理和Levin定理中的相关估计。
Sets with no Riesz bases of exponentials
本文证明了一类集合（包括平面中的圆盘和三角形）不存在指数Riesz基，给出了否定性结论。
We study the basis property of systems of exponentials with frequencies belonging to 'simple quasicrystals'. We show that a diophantine condition is necessary and sufficient for such a system to be a Riesz basis in L^2 on a finite union of intervals. For the proof we extend to BMO a theorem of Kesten about the discrepancy of irrational rotations of the circle.
We prove the existence of Riesz bases of exponentials of L^2(Omega), provided that Omega in R^d is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case.
Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $φ\in L^2(0,1)$. More precisely, we study when the system $\{φ(nx)\}_n$ is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space $\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.
We prove that every finite union of rectangles in $R^d$ admits a Riesz basis of exponentials.
We consider exponential systems $E\left(Λ\right)=\left\{ e^{iλt}\right\} _{λ\inΛ}$ for $Λ\subset\mathbb{Z}$. It has been shown by Londner and Olevskii in [9] that there exists a subset of the circle, of positive Lebesgue measure, so that every set Λwhich contains, for arbitrarily large N, an arithmetic progressions of length N and step $\ell=O\left(N^α\right)$, $α<1$, cannot be a Riesz sequence in the $L^{2}$ space over that set. On the other hand, every set admits a Riesz sequence containing arbitrarily long arithmetic progressions of length N and step $\ell=O\left(N\right)$. In this paper we…
Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$ endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $Δ=-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_iΔ^{-1/2}$ and $S_i=Δ^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second order Riesz transforms $T_{ij}=X_iΔ^{-1}X_j$ are bounded from $H^1$ …
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. Furthermor…
We show that any finite union of intervals supports a Riesz basis of exponentials
We provide a necessary and sufficient condition to ensure that a multi-tile $Ω$ of $R^d$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $ L^{2}(Ω)$. New examples are given and this characterization is generalized to abstract locally compact abelian groups.
Exponential frames and Riesz sequences at the critical density
本文研究了在临界密度下指数系统形成框架或里斯序列的条件，并证明了重复点集在该密度下自动成为里斯基底。
Suppose $Ω\subseteq\RR^d$ is a bounded and measurable set and $Λ\subseteq \RR^d$ is a lattice. Suppose also that $Ω$ tiles multiply, at level $k$, when translated at the locations $Λ$. This means that the $Λ$-translates of $Ω$ cover almost every point of $\RR^d$ exactly $k$ times. We show here that there is a set of exponentials $\exp(2πi t\cdot x)$, $t\in T$, where $T$ is some countable subset of $\RR^d$, which forms a Riesz basis of $L^2(Ω)$. This result was recently proved by Grepstad and Lev under the extra assumption that $Ω$ has boundary of measure 0, using methods from the theory of qua…
We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist pairwise disjoint sets $Λ_\ell \subset \mathbb{Z}$, $\ell=1, \ldots, L$, such that for every $J \subset \{ 1, \ldots , L \}$, the system $\{e^{2πi λx} : λ\in \cup_{\ell \in J} \, Λ_\ell \}$ is a Riesz basis for $L^2 ( \cup_{\ell \in J} \, I_\ell)$. Also, we show that for any disjoint intervals $I_\ell$, $\ell=1, \ldots, L$, contained in $[1,N)$ with $N \in \math…
Exponential Riesz bases in $L^2$ on two interval
本文研究了两个区间上指数系统成为里斯基基的充分条件，并揭示了此类系统可能存在的一个额外点效应。
Let $S$ be the union of finitely many disjoint intervals on the real line. Suppose that there are two real numbers $α, β$ such that the length of each interval belongs to $Z α+ Z β$. We use quasicrystals to construct a discrete set of real frequencies such that the corresponding system of exponentials is a Riesz basis in the space $L^2(S)$.
In this paper we study the $s$-dimensional Riesz transform of a finite measure $μ$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $μ$ implies that a nonlinear potential of exponential type is finite $μ$-almost everywhere. It appears to be the first result of this type for $s>1$.
We consider a regular indefinite Sturm-Liouville eigenvalue problem \{$-f" + q f = λr f$} on $[a,b]$ subject to general self-adjoint boundary conditions and with a weight function $r$ which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions $r$ for which the Riesz basis property can be completely characterized in terms of the local behavior of $r$ in a neighborhood of the …
Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix $A$ associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of $A$ in the phase space.
It is known that if $Ω\subset \mathbb{R}^{d}$ belongs to a class of multi-tiling domains when translated by a lattice $Λ$, there exists a Riesz basis of exponentials for $L^{2}(Ω)$ constructed using $k$ translates of the dual lattice $Λ^*$. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in prior work to this setting.
Weak type $(1, 1)$ of Riesz transform on some direct product manifolds with exponential volume growth
本文研究了直积流形 ${\mathbb{H}}^n \times M$ 上的 Riesz 变换，在体积加倍和热核上估计条件下建立了其弱型 $(1,1)$ 性质，并推广至热极大算子及更广泛的指数体积增长流形。
Under the assumption that $V \in L^2([0,π]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators $-d^2/dx^2+V$ in $L^2([0,π]; dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $-d^2/dx^2+V$ in $L^p([0,π]; dx)$, $p \in (1,\infty)$.
We establish a criterion for a set of eigenfunctions of the one-dimensional Schrödinger operator with distributional potentials and boundary conditions containing the eigenvalue parameter to be a Riesz basis for $\mathscr{L}_2(0,π)$.
Despite the recent advances in the theory of exponential Riesz bases, it is yet unknown whether there exists a set $S \subset \mathbb{R}^d$ which does not admit a Riesz spectrum, meaning that for every $Λ\subset \mathbb{R}^d$ the set of exponentials $e^{2πi λ\cdot x}$ with $λ\inΛ$ is not a Riesz basis for $L^2(S)$. As a meaningful step towards finding such a set, we construct a set $S \subset [-\frac{1}{2}, \frac{1}{2}]$ which does not admit a Riesz spectrum containing a nonempty periodic set with period belonging in $α\mathbb{Q}_+$ for any fixed constant $α> 0$, where $\mathbb{Q}_+$ denotes t…
Riesz bases of exponentials for convex polytopes with symmetric faces
本文证明对任意中心对称且各维面均中心对称的凸多面体，其 $L^2$ 空间中存在指数函数Riesz基，该结果在 $d>1$ 的所有维度上均为首次建立。
In 1996, H. Volkmer observed that the inequality \[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \] is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem \[ -y"=λ\, r(x)y,\quad y(-1)=y(1)=0 \] form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. Here the weight $r\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$. We present two criteria in terms of Weyl-Titchmarsh $m$-functions for the Volkmer inequality to be valid. Using these res…
Applying the solution to the Kadison-Singer problem, we show that every subset $\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\left\{ e^{iλx}\right\} _{λ\in Λ}$ such that $Λ\subset\mathbb{Z}$ is a set with gaps between consecutive elements bounded by ${\displaystyle \frac{C}{\left|\mathcal{S}\right|}}$. In the case when $\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $Λ$ can be deterministically constructed.
Let S be a bounded, Riemann measurable set in R^d, and L be a lattice. By a theorem of Fuglede, if S tiles R^d with translation set L, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles R^d with translation set L, S has a Riesz basis of exponentials. The proof is based on Meyer's quasicrystals.
We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.
Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has such a basis, removing interest in the existence question in this context. We then define tightness quantities for subsets to measure the conditioning of Riesz bases; for normalized tightness quantities, a value of one corresponds to an orthogonal basis, and a value of infinity corresponds to nonexistence of a basis. As an application, we obtain new weak evide…
Given a set $\mathcal{S}$ of positive measure on the circle and a set of integers $Λ$, one may consider the family of exponentials $E\left(Λ\right):=\left\{ e^{iλt}\right\}_{λ\inΛ}$ and ask whether it is a Riesz sequence in the space $L^{2}\left(\mathcal{S}\right)$. We focus on this question in connection with some arithmetic properties of the set of frequencies. Improving a result of Bownik and Speegle, we construct a set $\mathcal{S}$ such that $E\left(Λ\right)$ is never a Riesz sequence if $Λ$ contains arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N^{1-\varepsil…
We consider systems of exponentials with frequencies belonging to simple quasicrystals in $\mathbb{R}^d$. We ask if there exist domains $S$ in $\mathbb{R}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several …
There are no Riesz bases of exponentials in balls and triangles
本文证明在球和非退化三角形区域上，$L^2$空间不存在指数的Riesz基，并推广到具有光滑边界的凸集，方法是通过构造平稳系统与投影比较族导出矛盾。
Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,π]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We describe classes of complex potentials $v(x)= \sum_{2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v) $ contains a Riesz basis if and only …
We bound an exponential sum that appears in the study of irregularities of distribution (the low-frequency Fourier energy of the sum of several Dirac measures) by geometric quantities: a special case is that for all $\left\{ x_1, \dots, x_N\right\} \subset \mathbb{T}^2$, $X \geq 1$ and a universal $c>0$ $$ \sum_{i,j=1}^{N}{ \frac{X^2}{1 + X^4 \|x_i -x_j\|^4}} \lesssim \sum_{k \in \mathbb{Z}^2 \atop \|k\| \leq X}{ \left| \sum_{n=1}^{N}{ e^{2 πi \left\langle k, x_n \right\rangle}}\right|^2} \lesssim \sum_{i,j=1}^{N}{ X^2 e^{-c X^2\|x_i -x_j\|^2}}.$$ Since this exponential sum is intimately tied …
Let $\mathcal G$ be a stratified Lie group and $\{\X_j\}_{1 \leq j \leq n}$ a basis for the left-invariant vector fields of degree one on $\mathcal G$. Let $Δ= \sum_{j = 1}^n \X_j^2 $ be the sub-Laplacian on $\mathcal G$. The $j^{\mathrm{th}}$ Riesz transform on $\mathcal G$ is defined by $R_j:= \X_j (-Δ)^{-\frac{1}{2}}$, $1 \leq j \leq n$. In this paper, we provide a concrete construction of the "twisted truncated sector" which is related to the pointwise lower bound of the kernel of $R_j$ on $\mathcal G$. Then we obtain the characterisation of compactness of the commutators of $R_j$ with a f…
Concerning the stability of exponential systems and Fourier matrices
本文建立并系统研究 $\mathbb{R}^d$ 立方体并集上指数系统与一般 Fourier 矩阵极端奇异值之间的灵活联系，据此推导 Fourier 矩阵的新稳定性结果。
Basisness of Fucik eigenfunctions for the Dirichlet Laplacian
本文改进了Fucik特征值序列的充分条件，证明其对应的特征函数在$ L^2(0,π) $中构成Riesz基。
Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice
该研究针对频率限制在指定格点的情形，推导了平行六面体存在指数Riesz基的两个充分条件，分别推广了正交基情形的充要条件、通过生成矩阵谱范数给出约束。
Let $S\subset\R^d$ be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to $S$, $PW_S$, is defined to be the set of all square-integrable functions on $\R^d$ whose Fourier transforms vanish outside $S$. A sequence $(x_j:j\kin\N)$ in $\R^d$ is said to be a Riesz-basis sequence for $L_2(S)$ (equivalently, a complete interpolating sequence for $PW_S$) if the sequence $(e^{-i\la x_j,\cdot\ra}:j\kin\N)$ of exponential functions forms a Riesz basis for $L_2(S)$. Let $(x_j:j\kin\N)$ be a Riesz-basis sequence for $L_2(S)$. Given $λ>0$ and $f\in PW_S$, there is a unique…
Basisness and completeness of Fucik eigenfunctions for the Neumann Laplacian
研究Neumann拉普拉斯算子Fucik特征函数的基性质，证明其在$ L^2(0,π) $中的完备性及Riesz基性质，并描述双正交系统。
The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if $S_1, \ldots, S_K \subset [0,1]$ are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of $\mathbb{Z}$ into sets $Λ_1, \ldots, Λ_K$ such that $\bigcup_{k=1}^K \{ e^{2πi λ(\cdot)} χ_{S_k} : λ\in Λ_k \}$ is a Riesz bas…
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