arXiv subjects

Rachid Zarouf

Publications and source records attributed to Rachid Zarouf.

At least 19 recordsLinked to original sources

arXiv2026

Schäffer's matrix inequality: the exact asymptotic constant

Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Schäffer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subsequent work showed that $S_n$ grows like $\sqrt n$, but the sharp asymptotic constant has remained open for more than five decades. We resolve this problem by proving \[ \lim_{n\to\infty}\frac{S_n}{\sqrt n}=\sqrt e. \] Thus Schäffer's upper bound is asymptotically sharp, including its constant. Our proof is constructive, providing explicit Banach-space norms through duality and explicit matrices through the theory of model operators. At the analytic core of the argument, an extremal formulation of Schäffer's problem in the Wiener algebra reduces the matching asymptotic lower bound for $S_n$ to uniformly controlling the Taylor coefficients of products $QB_n$, where $B_n$ is a finite Blaschke product of degree $n$ and $Q$ is a polynomial factor. We optimize simultaneously the zero distribution of $B_n$ and the choice of $Q$. The resulting zeros follow a logarithmic asymptotic distribution, and a sharp uniform asymptotic analysis of the Taylor coefficients of $QB_n$ yields the constant $\sqrt e$. The corresponding model operators then yield matrices with these spectra that asymptotically attain Schäffer's bound.

math.FA↗

arXiv2026

Projected Inner-Function Dynamics and Crystalline Measures of Meyer-Blaschke type

We generalize a construction of Yves Meyer of sparse crystalline measures arising from powers of a Blaschke factor. Starting from a recursion $f_n=θ^n f_0$ on the unit circle where $θ$ is an inner function, we project the Fourier coefficient array $\widehat{f_n}(k)$ to the real line by placing its entries at the frequencies $k+αn$. We identify the role of model spaces in this construction: in Meyer's one-factor Blaschke recursion, the requirement that the coefficient array $\widehat{f_n}(k)$ vanish whenever $kn<0$ is equivalent to $f_0\in K_{zb_λ}$, and for general inner functions the condition $f_0\in K_{zθ}$ yields a purely atomic Radon measure with locally finite support and polynomial growth on the Fourier side. We also show that, when $f_0$ is holomorphic in an annulus containing the unit circle, exponential Fourier decay is sufficient to obtain a purely atomic Radon measure of polynomial growth, though not necessarily locally finite support. For finite Blaschke products, the coefficient recursion gives an explicit annihilating exponential polynomial whose zero set controls the support and separation of the inverse Fourier transform. This yields Meyer-Blaschke-type crystalline measures and Poisson identities with sampling and finite-truncation consequences.

math.CA↗

arXiv2026

Lower bounds in $H^2$-rational approximation to Blaschke products

We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose zeros are bounded away from the circle. The latter case depends on Fourier coefficients estimates for Blaschke products which are of independent interest.

math.CA↗

arXiv2026

Asymptotic sharpness of a Nikolskii type inequality for rational functions in the Wiener algebra

We establish the asymptotic sharpness of a Nikolskii type inequality proved by A. Baranov and R. Zarouf for rational functions $f$ in the Wiener algebra of absolutely convergent Fourier series, with at most $n$ poles, all lying outside the dilated disc $\frac{1}λ\mathbb{D}$, where $\mathbb{D}$ denotes the open unit disc and $λ\in[0,1)$ is fixed. More precisely, this inequality tells that the Wiener norm of such functions is bounded by their $H^{2}$-norm -- i.e., their norm in the Hardy space of the disc -- times a factor of order $\sqrt{\frac{n}{1-λ}}$. In this paper, we construct explicit test functions showing that this bound cannot be improved in general: the inequality is asymptotically sharp as $n\to\infty$, up to a universal constant, for every fixed $λ\in[0,1)$.

math.CA↗

arXiv2026

Simultaneous polynomial approximation in Beurling-Sobolev spaces via Blaschke products

Assuming that $ϕ(t)=o(t^2)$ as $t\to0$, we establish a lemma on simultaneous polynomial approximation in Orlicz-Beurling-Sobolev spaces $\ell_a^ϕ$. These spaces, endowed with the Luxemburg norm $\Vert \cdot \Vert_{\ell^ϕ}$, generalize the classical Beurling-Sobolev spaces $\ell_a^p$ for $p>2$. More precisely, we prove that for every $\varepsilon>0$, every $v\in\mathbb{N}$ and every function $φ$ continuous on $\partial\mathbb{D}$, there exist a polynomial $P(z)=\sum_{k=v}^d a_k z^k$ and a compact set $K\subset\partial\mathbb{D}$ with $m(K)>1-\varepsilon$ such that \[\|P\|_{\ell^ϕ}\le\varepsilon \quad \text{and}\quad \|P-φ\|_K\le\varepsilon.\] The proof relies on a result of independent interest describing the asymptotic behaviour of the Luxemburg norm $\|B^k\|_{\ell^ϕ}$ of powers of a finite Blaschke product $B$ which is not a monomial. This behaviour is governed by the comparison between $ϕ(t)$ and $t^2$ near $0$: the norms remain bounded when $ϕ\asymp t^2$, tend to $0$ when $ϕ=o(t^2)$, and diverge to $+\infty$ when $t^2=o(ϕ(t))$. A key ingredient in the proof is the qualitative limit $\sup_{j\ge0}|\widehat{B^k}(j)|\to0$ as $k\to\infty$. As an application of the simultaneous approximation lemma, we derive the existence of functions in $\ell_a^ϕ$ with universal properties, including Menshov universality of Taylor partial sums and universality with respect to radial boundary limits.

math.CV↗

arXiv2024

On the Exact Maxwell evolution equation of resonator dynamics

In a recent publication [Opt. Express 32, 20904 (2024)], the accuracy of the main evolution equation that governs resonator dynamics in the coupled-mode theory (CMT) was questioned. The study concluded that the driving force is proportional to the temporal derivative of the excitation field rather than the excitation field itself. This conclusion was reached with a derivation of an "exact" Maxwell evolution (EME) equation obtained directly from Maxwell's equations, which was further supported by extensive numerical tests. Hereafter, we argue that the original derivation lacks mathematical rigor. We present a direct and rigorous derivation that establishes a solid mathematical foundation for the EME equation. This new approach clarifies the origin of the temporal derivative in the excitation term of CMT and elucidates the approximations present in the classical CMT evolution equation through a straightforward argument.

physics.optics↗

arXiv2024

Bernstein-type inequalities for mean $n$-valent functions

We derive new integral estimates of the derivatives of mean $n$-valent functions in the unit disk. Our results develop and complement estimates obtained by E.P. Dolzhenko and A.A. Pekarskii, as well as recent inequalities obtained by the authors. As an application, we improve some inverse theorems of rational approximation due to Dolzhenko.

math.CV↗

arXiv2024

Analytic capacities in Besov spaces

We derive new estimates on analytic capacities of finite sequences in the unit disc in Besov spaces with zero smoothness, which sharpen the estimates obtained by N.K.Nikolski in 2005 and, for a range of parameters, are optimal. The work is motivated both from the perspective of complex analysis by the description of sets of zeros/uniqueness, and from the one of matrix analysis/operator theory by estimates on norms of inverses.

math.CV↗

arXiv2024

Bloch functions with wild boundary behaviour in $\mathbb{C}^N$

We prove the existence of functions $f$ in the Bloch space of the unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ with the property that, given any measurable function $φ$ on the unit sphere $\mathbb{S}_N$, there exists a sequence $(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in \mathbb{B}_N$, $$f(r_n(ζ-w)+w) \to φ(ζ)\text{ as }n\to \infty\text{, for almost every }ζ\in \mathbb{S}_N.$$ The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.

math.CV↗

arXiv2024

Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

In this work we investigate an inverse coefficient problem for the one-dimensional subdiffusion model, which involves a Caputo fractional derivative in time. The inverse problem is to determine two coefficients and multiple parameters (the order, and length of the interval) from one pair of lateral Cauchy data. The lateral Cauchy data are given on disjoint sets in time with a single excitation and the measurement is made on a time sequence located outside the support of the excitation. We prove two uniqueness results for different lateral Cauchy data. The analysis is based on the solution representation, analyticity of the observation and a refined version of inverse Sturm-Liouville theory due to Sini [35]. Our results heavily exploit the memory effect of fractional diffusion for the unique recovery of the coefficients in the model. Several numerical experiments are also presented to complement the analysis.

math.AP↗

arXiv2024

On the Fourier coefficients of powers of a finite Blaschke product

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell^{\infty}$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer's question on norms of inverses.

math.CV↗

arXiv2022

On the asymptotic behavior of Jacobi polynomials with first varying parameter

We investigate the large $n$ behavior of Jacobi polynomials with varying parameters $P_{n}^{(an+α,\,bn+β)}(1-2λ^{2})$ for $a,b >-1$ and $λ\in(0,\,1)$. This is a well-studied topic in the literature but some of the published results appear to be discordant. To address this issue we provide an in-depth investigation of the case $b = 0$, which is most relevant for our applications. Our approach is based on a new and surprisingly simple representation of $P_{n}^{(an+α,\,β)}(1-2λ^{2}),\:a>-1$ in terms of two integrals. The integrals' asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. As a consequence we prove that if $a\in(\frac{2λ}{1-λ},\infty)$ then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ shows exponential decay and we derive simple exponential upper bounds in this region. If $a\in(\frac{-2λ}{1+λ},\,\frac{2λ}{1-λ})$ then the decay of $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ is $\mathcal{O}(n^{-1/2})$ and if $a\in\{\frac{-2λ}{1+λ},\,\frac{2λ}{1-λ}\}$ then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ decays as $\mathcal{O}(n^{-1/3})$. A new phenomenon occurs in the parameter range $a\in(-1,\frac{-2λ}{1+λ})$, where we find that the behavior depends on whether or not $an+α$ is an integer: If $a\in(-1,\frac{-2λ}{1+λ})$ and $an+α$ is an integer then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ decays exponentially. If $a\in(-1,\frac{-2λ}{1+λ})$ and $an+α$ is not an integer then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ may increase exponentially depending on the proximity of the sequence $(an + α)_n$ to integers.

math.CA↗

arXiv2021

On the Fourier coefficients of powers of a Blaschke factor and strongly annular fonctions

We compute asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_λ^n$, where $b_λ(z)=\frac{z-λ}{1-λz}$ is the Blaschke factor associated to $λ\in\mathbb{D}$, $k\in[0,\infty)$ and $n$ is a large integer. We distinguish several regions of different asymptotic behavior of those coefficients in terms of $k$ and $n$. Given $β\in((1-λ)/(1+λ),(1+λ)/(1-λ))$ their decay is oscillatory for $k\in[βn,n/β]$. Given $α\in(0,(1-λ)/(1+λ))$ their decay is exponential for $k\in[0,nα]\cup[n/α,\infty).$ Airy-type behavior is happening near the $k$-transition points $n(1-λ)/(1+λ)$ and $n(1+λ)/(1-λ)$. The asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_λ^{n}$ are derived using standard tools of asymptotic analysis of Laplace-type integrals. More precisely, the integral defining the $k^{\rm th}$ Fourier coefficient of $b_λ^n$ is perfectly suited for an application of the method of stationary phase when $k\in\left(n(1-λ)/(1+λ),n(1+λ)/(1-λ)\right)$ and requires the use of the method of the steepest descent when $k\notin[n(1-λ)/(1+λ),n(1+λ)/(1-λ)]$. Uniform versions of those standard methods are required when $k$ approaches one of the boundaries $n(1-λ)/(1+λ),$ $n(1+λ)/(1-λ)$. As an application, we construct strongly annular functions with Taylor coefficients satisfying sharp summation properties.

math.CV↗

arXiv2020

Interpolation without commutants

We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carathéodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting approach of D. Sarason and B. Nagy-C.Foiaş. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This Functional extensions lemma plays a similar role as Sarason's Commutant lifting theorem but it only involves the predual of X and no Hilbert space structure is needed. As an example, we present the respective Pick-type interpolation theorems for Beurling-Sobolev spaces.

math.FA↗

arXiv2019

On Bernstein's inequality for polynomials

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $L^p-norm$.

math.CA↗

arXiv2019

$H^{\infty}$ interpolation and embedding theorems for rational functions

We consider a Nevanlinna-Pick interpolation problem on finite sequences of the unit disc D constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of H $\infty$ into Hardy or radial-weighted Bergman spaces in D is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constants. Mathematics Subject Classification (2010). Primary 15A60, 32A36, 26A33; Secondary 30D55, 26C15, 41A10.

math.CV↗

arXiv2017

A constructive approach to Schaeffer's conjecture

J.J. Schaeffer proved that for $any$ induced matrix norm and $any$ invertible $T=T(n)$ the inequality \[\left|\det T\right|\left\Vert T^{-1}\right\Vert \leq\mathcal{S}\left\Vert T\right\Vert ^{n-1}\] holds with $\mathcal{S}=\mathcal{S}(n)\leq\sqrt{en}$. He conjectured that the best $\mathcal{S}$ was actually bounded. This was rebutted by Gluskin-Meyer-Pajor and subsequent contributions by J. Bourgain and H. Queffelec that successively improved lower estimates on $\mathcal{S}$. These articles rely on a link to the theory of power sums of complex numbers. A probabilistic or number theoretic analysis of such inequalities is employed to prove the existence of $T$ with growing $\mathcal{S}$ but the explicit construction of such $T$ remains an open task. In this article we propose a constructive approach to Schaeffer's conjecture that is not related to power sum theory. As a consequence we present an explicit sequence of Toeplitz matrices with singleton spectrum $\{λ\}\subset\mathbb{D}-\{0\}$ such that $\mathcal{S}\geq c(λ)\sqrt{n}$. Our framework naturally extends to provide lower estimates on the resolvent $\left\Vert (ζ-T)^{-1}\right\Vert$ when $ζ\neq0$. We also obtain new upper estimates on the resolvent when the spectrum is given. This yields new upper bounds on $\left\Vert T^{-1}\right\Vert$ in terms of the eigenvalues of $T$ which slightly refine Schaeffer's original estimate.

math.NA↗

arXiv2017

$l_{p}$-norms of Fourier coefficients of powers of a Blaschke factor

We determine the asymptotic behavior of the $l_{p}$-norms of the sequence of Taylor coefficients of $b^{n}$, where $b=\frac{z-λ}{1-\barλz}$ is an automorphism of the unit disk, $p\in[1,\infty]$, and $n$ is large. It is known that in the parameter range $p\in[1,2]$ a sharp upper bound \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{2-p}{2p}} \end{align*} holds. In this article we find that this estimate is valid even when $p\in[1,4)$. We prove that \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{4}^A}\leq C_{4}\left(\frac{\log n}{n}\right)^{\frac{1}{4}} \end{align*} and for $p\in(4,\infty]$ that \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{1-p}{3p}} & . \end{align*} We prove that our upper bounds are sharp as $n$ tends to $\infty$ i.e. they have the correct asymptotic $n$ dependence.

math.CA↗