Related papers: On complete and incomplete exponential systems

We prove that any incomplete system of complex exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-\pi,\pi)$ is a subset of some complete and minimal system of exponentials. In addition, we prove analogous statement for systems of reproducing…

Given a relatively compact set $\Omega \subseteq \mathbb{R}$ of Lebesgue measure $|\Omega|$ and $\varepsilon > 0$, we show the existence of a set $\Lambda \subseteq \mathbb{R}$ of uniform density $D (\Lambda) \leq (1+\varepsilon) |\Omega|$…

Suppose that $\{\lambda_n\}_{n=1}^{\infty}$ is a sequence of distinct positive real numbers satisfying the conditions inf$\{\lambda_{n+1}-\lambda_n \}>0,$ and $\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty.$ We prove that the exponential system…

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a one-dimensional…

Let $\{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}$ be an exponential Schauder Basis for $L^2 (0,1)$, for $\lambda_n\in\mathbb{R}$, and let $\{r_n(t)\}_{n\in\mathbb{Z}}$ be its dual Schauder Basis. Let $A$ be a non-empty subset of the integers…

We prove that for a weight $w$, which has at least polynomial decay, there exists a complete and minimal system $\{e^{i\lambda_n t}\}_{n\in \mathbb{N}}$ of exponentials in weighted space $L^2(w)$ on $(-\pi,\pi)$, which is not hereditarily…

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $E(\Lambda,\varphi) = \{e^{2\pi i \lambda\cdot\varphi(x)}: \lambda\in\Lambda\}$ where the phase function $\varphi$ is a Borel measurable which…

A bounded measurable set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers ("frequencies") such that the exponential functions $e_\lambda(x) = \exp(2\pi i \lambda x)$,…

We study the properties of a system biorthogonal to a complete and minimal system of exponentials in $L^2(E)$, where $E$ is a finite union of intervals, and show that in the case when $E$ is a union of two or three intervals the…

Given discrete subsets $\Lambda_j\subset {\Bbb R}^d$, $j=1,...,q$, consider the set of windowed exponentials $\bigcup_{j=1}^{q}\{g_j(x)e^{2\pi i <\lambda,x>}: \lambda\in\Lambda_j\}$ on $L^2(\Omega)$. We show that a necessary and sufficient…

Let $d$ be a positive integer, and let $\mu$ be a finite measure on $\br^d$. In this paper we ask when it is possible to find a subset $\Lambda$ in $\br^d$ such that the corresponding complex exponential functions $e_\lambda$ indexed by…

Let $A$ be a countable and discrete subset of ${\Bbb R}^d$, $d \ge 2$, of positive upper Beurling density. Let $K$ denote a bounded symmetric convex set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It is known…

For $\sigma\in (0,+\infty)$, denote by $B_\sigma^{\infty}$ the Bernstein space (of type $\sigma$) of all entire functions of exponential type $\leq \sigma$ bounded on real axis $\R$. Let $I_d\subset \R$ be a segment of length $d>0$. We…

For every set $S$ of finite measure in $\mathbb{R}$ we construct a discrete set of real frequencies $\Lambda$ such that the exponential system $\{\exp(i\lambda t),\lambda\in\Lambda\}$ is a frame in $L^2(S)$

We establish a criterion for the completeness of an exponential system in the spaces of functions continuous on a convex compact set and holomorphic in the interior of this compact set, as well as in the spaces of holomorphic functions in…

We are concerned with an harmonic analysis in Hilbert spaces $L^2(\mu)$, where $\mu$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_\lambda(x) = \exp(2\pi i…

Let $\mu$ be a finite positive measure on the real line. For $a>0$ denote by $\EE_a$ the family of exponential functions $$\EE_a=\{e^{ist}| \ s\in[0,a]\}.$$ The exponential type of $\mu$ is the infimum of all numbers $a$ such that the…

A uniformly bounded complete orthonormal system of functions $\Theta =\{ \theta_n\}_{n=1}^{\infty},$ $ \|\theta_n\|_{L^\infty_{[0,1]} } \leq M $ is constructed such that $\sum_{n=1}^{\infty} a_{n}\theta_{n}$ converges almost everywhere on…

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…

We present new completeness conditions for exponential systems on the complex plane in Banach algebras of continuous functions on a compact with a connected complement that are simultaneously holomorphic in the interior of this compact if…