arXiv subjects
Publications and source records attributed to Karim Kellay.
We study the zeros of functions in the Dirichlet space. Using extremal functions, we produce a necessary and sufficient condition for a sequence of points in the unit disk to be a zero set of the classical Dirichlet space. This Shapiro-Shields type condition involves kernels of the harmonic Dirichlet space associated with measures depending on the zero sequence.
We obtain a characterization of complete interpolating sequences in a class of Fock-type spaces with radial weights for which such sequences exist. Our criterion is formulated in terms of logarithmic separation and controlled perturbations of a reference sequence satisfying an Avdonin-type condition. This provides a geometric description of complete interpolating sequences and extends previous results of Borichev--Lyubarskii and Baranov--Belov--Borichev on Riesz bases of reproducing kernels in Fock-type spaces. It also yields explicit density criteria for sampling and interpolating sequences.
We study the local Dirichlet integral of distance functions and their behavior within the harmonic Dirichlet space. We provide estimates for the local Dirichlet integral of distance functions, which allow us to study their membership in the algebra of multipliers of the Dirichlet space. We give sufficient condition for a closed subset of the unit circle to be polar and we also examine cyclicity in the harmonic Dirichlet spaces.
We establish a general form of Wiener's lemma for measures on locally compact abelian (LCA) groups by using Fourier analysis and the theory of F{ø}lner sequences. Our approach provides a unified framework that that encompasses both the discrete and continuous cases. We also show a version of Wiener's lemma for Bochner-Riesz means on both R^d and T^d . Mathematics Subject Classification (2010). 43A25.
In this paper we show that, if an increasing sequence $Λ=(λ_k)_{k\in\mathbb{Z}}$ has gaps going to infinity $λ_{k+1}-λ_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence $(a_k)_{k\in\mathbb{Z}}$ and every $N\geq 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=0}^N a_k e^{2iπλ_k t}\right|\,\mbox{d}t$$ further, if $\sum_{k\in\mathbb{Z}}\dfrac{1}{1+|λ_k|}<+\infty$,$$ B\max_{|k|\leq N}|a_k|\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=-N}^N a_k e^{2iπλ_k t}\right|\,\mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $Λ$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by Ingham for $T\geq 1$ under the condition that $λ_{k+1}-λ_k\geq 1$. The main novelty is that if those gaps go to infinity, then $T$ can be taken arbitrarily small. The result is new even when the $λ_k$'s are integers where it extends a result of McGehee, Pigno and Smith. The results are then applied to observability of Schrödinger equations with moving sensors.
Tolokonnikov's Corona Theorem is used to obtain two results on cyclicity in Besov-Dirichlet spaces.
The aim of this paper is to extend two results from the Paley--Wiener setting to more generalmodel spaces. The first one is an analogue of the oversampling Shannon sampling formula. The second one is a version of the Donoho--Logan Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley--Wiener space into an $L^2(\R,μ)$ space.
In this paper, we give a direct quantitative estimate of $L^1$norms of non-harmonic trigonometric polynomials over large enough intervals. This extends the result by Konyagin and Mc Gehee, Pigno, Smith to the settingof trigonometric polynomials with non-integer frequencies.The result is a quantitative extension of a result by Nazarov and also covers a resultby Hudson and Leckband when the length of the interval goes to infinity.
We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case p = 2 and the uniform case p = +$\infty$.
In the classical phase retrieval problem in the Paley-Wiener class $PW_L$ for $L>0$, i.e. to recover $f\in PW_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by $\mathfrak{F}_a f$ the resulting function. The operator $\mathfrak{F}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of $\mathfrak{F}_a$. We show that $\mathfrak{F}_af$ is no longer bandlimited but is still wide-banded. We then investigate the effect of $\mathfrak{F}_a$ on the stability of phase retrieval by estimating the quantity $\inf_{|c|=1}\|cf-\mathfrak{F}_af\|_2$. We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity $\inf_{|c|=1}\|c\mathfrak{F}_bf-\mathfrak{F}_af\|_2$. We show that this quantity is dominated by the distance between $a$ and $b$.
We discuss random interpolation in weighted Dirichlet spaces $\mathcal{D}_α$, $0\leq α\leq 1$. While conditions for deterministic interpolation in these spaces depend on capacities which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $α=1/2$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for $\mathcal{D}_α$ are exactly the almost sure separated sequences when $0\le α<1/2$ (which includes the Hardy space $H^2=\mathcal{D}_0$), and they are exactly the almost sure zero sequences for $\mathcal{D}_α$ when $1/2 \leq α\le 1$ (which includes the classical Dirichlet space $\mathcal{D}=\mathcal{D}_1$).
This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given f $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x|} dx), we find all functions g $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x| dx}), such that |f (x)| = |g(x)| for all x $\in$ R. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of f and g, and determine if these constraints force uniqueness of the solution.
Let $μ$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(μ)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. We give a sufficient condition on a Borel subset $E$ of the unit circle which ensures that $E$ is a uniqueness set for $\mathcal{D}(μ)$. {We also give somes examples of positive Borel measures $μ$ and uniqueness sets for $\mathcal{D}(μ)$.}
We give a simple proof of the fact that a finite measure $μ$ on the unit disk is a Carleson measure for the Dirichlet space if it satisfies the Carleson one-box condition $μ(S(I))=O(ϕ(|I|))$, where $ϕ:(0,2π]\to(0,\infty)$ is an increasing function such that $\int_0^{2π}(ϕ(x)/x)\,dx<\infty$. We further show that the integral condition on $ϕ$ is sharp.
We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e. those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP theoremfor model spaces associated with bounded derivative inner functions. Considering meromorphic inner functions allows us tointroduce a new geometric density condition, in terms of which the sampling sets are completely characterized. This, incomparison to Volberg's characterization of sampling measures in terms of harmonic measure, enables us to obtain explicitestimates on the sampling constants. The methods combine Baranov-Bernstein inequalities, reverse Carleson measures andRemez inequalities .
We study multiple sampling, interpolation and uniqueness for the classical Fock spaces in the case of unbounded multiplicities. We show that there are no sequences which are simultaneously sampling and interpolating when the multiplicities tend to infinity.
Let $μ$ be a positive finite measure on the unit circle and $\mathcal{D} (μ)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \mathcal{D} (μ)$ is cyclic if and only if $c\_μ(Z (f))= 0$, where $c\_μ$ is the capacity associated with $\mathcal{D} (μ)$ and $Z(f)$ is the zero set of $f$. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.
The harmonic Dirichlet space $\cal{D} (\mathbb{T})$ is the Hilbert space of functions $f \in L^2(\mathbb{T})$ such that $$\|f\|_{\cal{D} (\mathbb{T})}^2 := \sum_{n\in\mathbb{Z}} (1+|n|)|\hat{f}(n)|^2 < \infty.$$ We give sufficient conditions for $f$ to be cyclic in $\cal{D} (\mathbb{T})$, in other words, for $\{ζ^nf(ζ):\ n\geq 0\}$ to span a dense subspace of $\cal{D} (\mathbb{T})$.