arXiv subjects
Publications and source records attributed to Myriam Ounaïes.
Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le ρ$, where $ρ>1$. Cassels proved that, under an additional restriction on $ρ$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le \left(\frac{ρ^{2n}-1}{ρ^2-1}\right)^{\!n} \] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $ρ$. In this paper, we confirm Alexander's conjecture.
We consider inner functions $Θ$ with the zero set $\mathcal Z(Θ)$ such that the quotient algebra $H^\infty / ΘH^\infty$ satisfies the Strong Invertibility Property (SIP), that is for every $\varepsilon>0$ there exists $δ>0$ such that the conditions $f \in H^\infty$, $\|[f]\|_{H^\infty/ ΘH^\infty}=1$, $\inf_{\mathcal Z(Θ)} |f| \ge 1-δ$ imply that $[f]$ is invertible in $H^\infty / ΘH^\infty$ and $\| 1/ [f] \|_{H^\infty/ ΘH^\infty}\le 1+\varepsilon$. We prove that the SIP is equivalent to the maximal asymptotic growth of $Θ$ away from its zero set. We also describe inner functions satisfying the SIP in terms of the narrowness of their sublevel sets and relate the SIP to the Weak Embedding Property introduced by P.Gorkin, R.Mortini, and N.Nikolski as well as to inner functions whose Frostman shifts are Carleson--Newman Blaschke products. We finally study divisors of inner functions satisfying the SIP. We describe geometrically the zero set of inner functions such that all its divisors satisfy the SIP. We also prove that a closed subset $E$ of the unit circle is of finite entropy if and only if any singular inner function associated to a singular measure supported on $E$ is a divisor of an inner function satisfying the SIP.
We give a sharp bound for the Lebesgue constant associated to Leja sequences in the complex unit disk, confirming a conjecture made by Calvi and Phung in 2011
In a first (theoretical) part of this paper, we prove a number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture, building on ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne that were recently reinterpreted by Draisma and de Jong in terms of $p$-adic valuations. In a second (computational) part, we present ideas improving upon Diaz-Toca and Gonzalez-Vega's Gröbner basis approach to the Casas-Alvero conjecture. One application is an extension of the proof of Graf von Bothmer et al. to the cases $5p^k$, $6p^k$ and $7p^k$ (that is, for each of these cases, we elaborate the finite list of primes $p$ for which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree 12 (the smallest open case).
The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is bounded by $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\D^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust--Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\D^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{\log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.
The Sidon constant for the index set of nonzero m-homogeneous polynomials P in n complex variables is the supremum of the ratio between the l^1 norm of the coefficients of P and the supremum norm of P in D^n. We present an estimate which gives the right order of magnitude for this constant, modulo a factor depending exponentially on m. We use this result to show that the Bohr radius for the polydisc D^n is bounded from below by a constant times sqrt((log n)/n).
We show that a discrete sequence $Λ$ of the complex plane is the union of $n$ interpolating sequences for the Hörmander algebras $A_p$ if and only if the trace of $A_p$ on $Λ$ coincides with the space of functions on $Λ$ for which the divided differences of order $n-1$ are uniformly bounded. The analogous result holds in the unit disk for Korenblum-type algebras.
We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.