A classical result of Arne Beurling states that the Fourier transform of a nonzero complex Borel measure $μ$ on the real line cannot vanish on a set of positive Lebesgue measure if $μ$ has certain decay. We prove a several variable analogue of Beurling's theorem by exploring its connection with the well-known problem concerning the density of linear span of exponentials in a certain weighted normed linear space of continuous functions. In the process, we also prove some new results of this genre and establish an equivalence between the above two problems. We also obtain a generalisation of Beurling's theorem and prove these results on the $n$-dimensional torus $\mathbb{T}^n.$
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.
The classical completeness problem raised by Beurling and independently by Wintner asks for which $ψ\in L^2(0,1)$, the dilation system $\{ψ(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $ψ$ is identified with its extension to an odd $2$-periodic function on $\mathbb{R}$. This difficult problem is nowadays commonly called as the Periodic Dilation Completeness Problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over the infinite-dimensional pol…
We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $ 0 < a < 1.$ We show that such functions are entire vectors for the Schrödinger representations of the Heisenberg group. If an eigenfunction $f$ of the Fourier transform satisfies the above condition we show that the Hermite coefficients of $f$ have certain exponential decay which depends on $a$.
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\rightar…
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.
In this paper we investigate a high dimensional version of Selberg's minorant problem for the indicator function of an interval. In particular, we study the corresponding problem of minorizing the indicator function of the box $Q_{N}=[-1,1]^N$ by a function whose Fourier transform is supported in the same box $Q_N$. We show that when the dimension is sufficiently large there are no minorants with positive mass and we give an explicit lower bound for such dimension. On the other hand, we explicitly construct minorants for dimensions $1,2,3,4$ and $5$ and, as an application, we use them to produ…
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…
We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{iλ_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a one-dimensional defect. This means that there is at most one (up to a constant factor) function $f$ which is orthogonal to all the summands in its formal Fourier series $\sum_n (f,\tilde e_n) e^{iλ_n t}$, where $\{\tilde e_n\}$ is the system biorthogonal to $\{e^{iλ_n t}\}$. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential …
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…
For a locally finite point set $Λ\subset \mathbb{R}$, consider the collection of exponential functions given by $\mathcal{E}_Λ:= \{e^{i λx} : λ\in L \}$. We examine the question whether $\mathcal{E}_Λ$ spans the Hilbert space $L^2[-π,π]$, when $Λ$ is random. For several point processes of interest, this belongs to a certain critical case of the corresponding question for deterministic $Λ$, about which little is known. For $Λ$ the continuum sine kernel process, obtained as the bulk limit of GUE eigenvalues, we establish that $\mathcal{E}_Λ$ is indeed complete. We also answer an analogous questi…
Given an expansive matrix $R\in M_d({\mathbb Z})$ and a finite set of digit $B$ taken from $ {\mathbb Z}^d/R({\mathbb Z}^d)$. It was shown previously that if we can find an $L$ such that $(R,B,L)$ forms a Hadamard triple, then the associated fractal self-affine measure generated by $(R,B)$ admits an exponential orthonormal basis of certain frequency set $Λ$, and hence it is termed as a spectral measure. In this paper, we show that if #$B<|\det (R)|$, not only it is spectral, we can also construct arbitrarily sparse spectrum $Λ$ in the sense that its Beurling dimension is zero.
We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains $ Ω\subset \mathbb R^d$. We obtain various sufficient conditions on the behavior of the coefficients near the boundary of $Ω$ which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, $-\frac{1}{ρ_\infty}\,\nabla\cdot ρ_\infty\mathbb D\nabla$. The proofs are based on the method developed in [29] for quantum confinement on bounded domains in $\mathbb R^d$. In particular for stochastic confinement we combine the Liouville prope…
Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type that vanishes on $ Z$ and satisfies the constraint $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$, $y\to \pm\infty$. Our results generalize and develop a joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions on the complex plane, as well as to the completeness…
We survey recent results concerning the hereditary completeness of some special systems of functions and the spectral synthesis problem for a related class of linear operators. We present a solution of the spectral synthesis problem for systems of exponentials in $L^2(-π, π)$. Analogous results are obtained for the systems of reproducing kernels in the de Branges spaces of entire functions. We also apply these results (via a functional model) to the spectral theory of rank one perturbations of compact self-adjoint operators.
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 …
