Related papers: Complementability of exponential systems

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…

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…

Given a bounded domain $\Omega \subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 \pi i x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for…

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 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…

Let $\Gamma$ be a subset of $\{0,1,2,...\}$. We show that if $\Gamma$ has `gaps' then the completeness and frame properties of the system $\{t^ke^{2\pi i nt}: n\in\mathbb{Z},k\in\Gamma\}$ differ from those of the classical exponential…

Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems…

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces $\mathcal{H}(E)$. Namely, we describe the de Branges spaces $\mathcal{H}(E)$ such that all $M$-bases of reproducing kernels (i.e., complete…

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…

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 $L^2(S)$

Let a sequence $\Lambda\subset\mathbb{C}$ be such that the corresponding system of exponential functions $\mathcal{E}(\Lambda):=\{e^{i\lambda t}\}_{\lambda\in\Lambda}$ is complete and minimal in $L^2(-\pi,\pi)$ and thus each function $f\in…

For a locally finite point set $\Lambda \subset \mathbb{R}$, consider the collection of exponential functions given by $\mathcal{E}_{\Lambda}:= \{e^{i \lambda x} : \lambda \in L \}$. We examine the question whether $\mathcal{E}_{\Lambda}$…

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…

The exponential kernel \[E{g}(\lambda,w) = \exp -\frac{1}{\pi}\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-\lambda) } da(u ),\] where the compactly supported bounded measurable function $g$ satisfies $0 \leq g\leq 1,$ and suitably…

It is shown that the complex field equipped with the "approximate exponential map", defined up to ambiguity from a small group, is quasiminimal: every automorphism-invariant subset of the field is countable or co-countable. If the ambiguity…

We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane.

Let $G$ be a second-countable amenable group with a uniform $k$-approximate lattice $\Lambda$. For a projective discrete series representation $(\pi, \mathcal{H}_{\pi})$ of $G$ of formal degree $d_{\pi} > 0$, we show that $D^-(\Lambda) \geq…

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 $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We show that the expansion of $\mathbb{T}$ by its natural exponential function is model complete and locally o-minimal. We give an axiomatization of the…

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…