arXiv subjects

Elias Zikkos

Publications and source records attributed to Elias Zikkos.

8 recordsLinked to original sources

arXiv2026

Hardy Subspaces with Sparse Fourier Spectrum and Muntz Space

Let $\Lambda=\{\lambda_n\}_{n=1}^{\infty}\subset\mathbb{N}$ with $\lambda_n$ strictly increasing and such that $\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty$. We show that a Hardy subspace $H^2 (\mathbb{D}, \Lambda)$ consisting of functions with sparse Fourier spectrum $\Lambda$ coincides with a M\"{u}ntz space $\overline{M^2_{\Lambda}}(\mathbb{D})$ characterized by square-summability of coefficients relative to a biorthogonal family. As consequences, we obtain a new characterization of the Hardy norm in $H^2 (\mathbb{D}, \Lambda)$ and an integral representation formula for the Fourier coefficients. The proof uses the biorthogonal representation developed in a previous work of the author.

math.FA↗

arXiv2026

The Gaussian Gabor system at the critical density is a weighted lower semi frame

We show that the Gaussian Gabor system at the critical density is a weighted lower semi frame, thus resolving an open question. This complements recent results showing that the system is neither a weighted frame nor admits a reproducing partner. In fact, after removing any atom from the system, suitable weights produce a complete Riesz Fischer sequence and lower semi frame.

math.FA↗

arXiv2025

On Markushevich bases $\{x^{λ_n}\}_{n=1}^{\infty}$ for their closed span in weighted $L^2 (A)$ spaces over sets $A\subset [0,\infty)$ of positive Lebesgue measure, hereditary completeness, and moment problems

Inspired by the work of Borwein and Erdelyi \cite{BE1997JAMS} on generalizations of Müntz's theorem, we investigate the properties of the system $\{x^{λ_n}\}_{n=1}^{\infty}$ in weighted $L^p (A)$ spaces, for $p\ge 1$, denoted by $L^p_w (A)$, where (I) $A$ is a measurable subset of the real half-line $[0,\infty)$ having positive Lebesgue measure, (II) $w$ is a non-negative integrable function defined on $A$, and (III) $\{λ_n\}_{n=1}^{\infty}$ is a strictly increasing sequence of positive real numbers such that $\inf\{λ_{n+1}-λ_n \}>0$ and $\sum_{n=1}^{\infty}λ_n^{-1}<\infty$. We prove that a function $f$ in $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ in the Hilbert space $L^2_w (A)$, admits the $\bf{Fourier-type}$ series representation $f(x)=\sum_{n=1}^{\infty} \langle f, r_n\rangle_{w,A} x^{λ_n}$ a.e on $A$, where $\{r_n\}_{n=1}^{\infty}$ is the unique biorthogonal family of $\{x^{λ_n}\}_{n=1}^{\infty}$ in $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ in $L^2_w (A)$. As a result, we show that the system $\{x^{λ_n}\}_{n=1}^{\infty}$ is a $\bf{Markushevich\,\, basis}$ for $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ in $L^2_w (A)$. Furthermore, we consider a $\bf{moment\,\, problem}$. Finally, if $m\le w(x)\le M$ on $A$ for some positive numbers $m$ and $M$ and the set $A$ contains an interval $[a, r_A]$, where $a\ge 0$ and $r_A$ is the essential supremum of $A$, we prove that the system $\{x^{λ_n}\}_{n=1}^{\infty}$ is $\bf{hereditarily\,\, complete}$ in $\overline{\text{span}}\{x^{λ_n}\}_{n=1}^{\infty}$ in the space $L^2_w(A)$. As a result, a general class of compact operators on the closure is constructed that admit spectral synthesis.

math.FA↗

arXiv2025

On Strong Markushevich bases $\{t^{λ_n}\}_{n=1}^{\infty}$ in their closed span in $L^2 (0, 1)$ and characterizing a subspace of $H^2 (\mathbb{D})$

Let $Λ=\{λ_n\}_{n=1}^{\infty}$ be a strictly increasing sequence of positive real numbers such that $\sum_{n=1}^{\infty}\frac{1}{λ_n}<\infty$ and $\inf(λ_{n+1}-λ_n)>0$. We investigate properties of the closed span of the system $\{t^{λ_n}\}_{n=1}^{\infty}$ in $L^2 (0,1)$, denoted by $\overline{M_Λ}$, and of the unique biorthogonal family $\{r_n (t)\}_{n=1}^{\infty}$ to the system $\{t^{λ_n}\}_{n=1}^{\infty}$ in $\overline{M_Λ}$. We show that the system $\{t^{λ_n}\}_{n=1}^{\infty}$ is a strong Markushevich basis in $\overline{M_Λ}$ and we obtain a series representation for functions in $\overline{M_Λ}$. We also construct a general class of operators on $\overline{M_Λ}$ that admit spectral synthesis. In particular, for all $ρ\in (0,1)$ the operator $T_ρ(f)=f(ρx)$ on $\overline{M_Λ}$ admits spectral synthesis. In addition, we characterize a certain subspace of the classical Hardy space $H^2 (\mathbb{D})$. Under the extra assumption that $Λ\subset\mathbb{N}$, let $H^2(\mathbb{D}, Λ)$ consist of functions $f$ in $H^2(\mathbb{D})$ so that the Fourier coefficients $c_n$ of the boundary function $f(e^{iθ})$ vanish for all $n\notin Λ$. We prove that $f\in H^2(\mathbb{D}, Λ)$ if and only if $f\in\overline{M_Λ}$ and $\sum_{n=1}^{\infty}\left| \langle f, r_n\rangle \right|^2<\infty$, where $\langle f, g\rangle= \int_{0}^{1} f(t)\cdot \overline{g(t)}\, dt$.

math.FA↗

arXiv2024

On exact systems $\{t^α\cdot e^{2πi nt}\}_{n\in\mathbb{Z}\setminus A}$ in $L^2 (0,1)$ which are not Schauder Bases and their generalizations

Let $\{e^{iλ_n t}\}_{n\in\mathbb{Z}}$ be an exponential Schauder Basis for $L^2 (0,1)$, for $λ_n\in\mathbb{R}$, and let $\{r_n(t)\}_{n\in\mathbb{Z}}$ be its dual Schauder Basis. Let $A$ be a non-empty subset of the integers containing exactly $M$ elements. We prove that for $α>0$ the weighted system \[ \{t^α\cdot r_n(t)\}_{n\in\mathbb{Z}\setminus A} \] is exact in the space $L^2 (0,1)$, that is, it is complete and minimal in $L^2 (0,1)$, if and only if \[ M-\frac{1}{2}\le α< M+\frac{1}{2}. \] We also show that such a system is not a Riesz Basis for $L^2 (0,1)$. In particular, the weighted trigonometric system $\{t^α\cdot e^{2πi n t}\}_{n\in\mathbb{Z}\setminus A}$ is exact in $L^2 (0,1)$, if and only if $α\in [M-\frac{1}{2}, M+\frac{1}{2})$, but it is not a Schauder Basis for $L^2 (0,1)$.

math.FA↗

arXiv2024

Hereditary completeness of Exponential systems $\{e^{λ_n t}\}_{n=1}^{\infty}$ in their closed span in $L^2 (a, b)$ and Spectral Synthesis

Suppose that $\{λ_n\}_{n=1}^{\infty}$ is a sequence of distinct positive real numbers satisfying the conditions inf$\{λ_{n+1}-λ_n \}>0,$ and $\sum_{n=1}^{\infty}λ_n^{-1}<\infty.$ We prove that the exponential system $\{e^{λ_n t}\}_{n=1}^{\infty}$ is hereditarily complete in the closure of the subspace spanned by $\{e^{λ_n t}\}_{n=1}^{\infty}$ in the space $L^2 (a,b)$. We also give an example of a class of compact non-normal operators defined on this closure which admit spectral synthesis.

math.FA↗

arXiv2022

Characterizing Riesz Bases via Biorthogonal Riesz-Fischer sequences

In this note we prove that if two Riesz-Fischer sequences in a separable Hilbert space $H$ are biorthogonal and one of them is complete in $H$, then both sequences are Riesz bases for $H$. This complements a recent result by D. T. Stoeva where the same conclusion holds if one replaces the phrase ``Riesz-Fischer sequences'' by ``Bessel sequences''.

math.FA↗

arXiv2022

The closed span of some Exponential system $E_Λ$ in the spaces $L^p(γ,β)$, properties of a Biorthogonal family to $E_Λ$ in $L^2(γ,β)$, Moment problems, and a differential equation of Carleson

A set of complex numbers $Λ=\{λ_n,μ_n\}_{n=1}^{\infty}$ with multiple terms \[ \{λ_n,μ_n\}_{n=1}^{\infty}:= \{\underbrace{λ_1,λ_1,\dots,λ_1}_{μ_1 - times}, \underbrace{λ_2,λ_2,\dots,λ_2}_{μ_2 - times},\dots, \underbrace{λ_k,λ_k,\dots,λ_k}_{μ_k - times},\dots\} \] is said to belong to the $\bf ABC$ class if it satisfies three conditions: $\bf (A)$ $\sum_{n=1}^{\infty}μ_n/|λ_n|<\infty$, $\bf (B)$ $\sup_{n\in\mathbb{N}}|\argλ_n|<π/2$, $\bf (C)$ $Λ$ is an interpolating variety for the space of entire functions of exponential type zero. Assuming that $Λ\in\bf ABC$, we characterize in the spirit of the Müntz-Szász theorem, the closed span of its associated exponential system \[ E_Λ:=\{x^k e^{λ_n x}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,μ_n-1\} \] in the Banach spaces $L^p(γ,β)$, where $-\infty<γ<β<\infty$ and $p\ge 1$. Related to $E_Λ$, we explore the properties of its unique biorthogonal sequence \[ r_Λ=\{r_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots,μ_n-1\}\subset\overline{\text{span}}(E_Λ) \] in $L^2(γ,β)$. As a result, we find a solution to the Moment problem \[ \int_γ^β f(t)\cdot t^k e^{\overline{λ_n} t}\, dt=d_{n,k},\qquad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,\dots ,μ_n-1,\quad d_{n,k}=O(e^{a\Reλ_n})\,\, for\,\, a<β. \] Finally, we characterize the solution space of a differential equation of infinite order, studied by L. Carleson.

math.CA↗