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K. Kellay

Publications and source records attributed to K. Kellay.

7 recordsLinked to original sources
arXiv2019

Riesz bases of reproducing kernels in small Fock spaces

We give a complete characterization of Riesz bases of normalized reproducing kernels in the small Fock spaces $\mathcal{F}^2_φ$, the spaces of entire functions $f$ such that $f\mathrm{e}^{-φ} \in L^{2}(\mathbb{C})$, where $φ(z)= (\log^+|z|)^{β+1}$, $0< β\leq 1$.The first results in this direction are due to Borichev-Lyubarskii who showed that $φ$ with $β=1$ is the largest weight for which the corresponding Fock space admits Riesz bases of reproducing kernels. Later, such bases were characterized by Baranov-Dumont-Hartman-Kellay in the case when $β=1$. The present paper answers a question in Baranov et al. by extending their results for all parameters $β\in (0,1)$. Our results are analogous to those obtained for the case $β=1$ and those proved for Riesz bases of complex exponentials for the Paley-Wiener spaces. We also obtain a description of complete interpolating sequences in small Fock spaces with corresponding uniform norm.

math.CA↗
arXiv2015

Constructive approximation in de Branges-Rovnyak spaces

In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f\_r(z):=f(rz)~(r \textless{} 1)$. We show that this is \emph{not} the case for the de Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in $\cH(b)$. We give the first constructive proof of this fact.

math.FA↗
arXiv2014

Kernel estimate and capacity in Dirichlet type spaces

Let $μ$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(μ)$, associated to $μ$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. First, we give an estimate of the norm of the reproducing kernel $k^μ$ of $\mathcal{D}(μ)$. Next, we study the notion of $μ$-capacity associated to $\mathcal{D}(μ)$, in the sense of Beurling--Deny. Namely, we give an estimate of $μ$-capacity of arcs in terms of the norm of $k^μ$. We also provide a new condition on closed sets to be $μ$-polar. Note that in the particular case where $μ$ is the Lebesgue measure, this condition coincides with Carleson's condition \cite{Ca}. Our method is based on sharp estimates of norms of some outer test functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.

math.CV↗
arXiv2010

Level sets and Composition operators on the Dirichlet space

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.

math.FA↗
arXiv2008

Ideaux fermes d'algebres de Beurling analytiques sur le bidisque

We study the closed ideal in the Beurling algebras $\mathcal{A}^{+}_{α,β}$ of holomorphic function $f$ in the bidisc such that $\sum_{n,m\geq 0}|\hat{f}(n,m)|(1+n)^α(1+m)^β<+\infty$. We determine the function $f\in\mathcal{A}^{+}_{α,β}$ such that the ideals generated by $f$ coincide with the ideal generated by their zeros set.

math.CV↗