arXiv subjects

Adem Limani

Publications and source records attributed to Adem Limani.

18 recordsLinked to original sources

arXiv2026

The Uncertainty Principle in Harmonic Analysis -- Lecture Notes on Selected Topics

These lecture notes are devoted to selected topics related to the uncertainty principle in harmonic analysis. Rather than attempting a systematic treatment, we emphasize only a number of both classical and deep manifestations of this principle, mainly from the perspective of Fourier analysis on the unit circle and on the real line. We consider problems of uniqueness and reconstruction for Fourier series and Fourier transforms, the influence of spectral gaps, and the role of logarithmic integrability in questions of approximation and quasi-analyticity. Central results discussed include the Paley--Wiener theorem, the Beurling--Malliavin multiplier theorem, and the Ivashev--Musatov theorem. These notes are intended as an entry point toward the research literature, with several sections pointing in the direction of more recent developments.

math.CA↗

arXiv2026

Asymmetric uniqueness sets in $\ell^q$

We exhibit an asymmetry phenomenon for uniqueness sets in $\ell^q$. Specifically, we construct sets that do not support measures with $\ell^q$-summable Fourier coefficients, yet simultaneously support measures whose positive frequencies decay faster than polynomials. In the language of Fourier uniqueness, this highlights a striking divergence between the unilateral and bilateral $\ell^q$ uniqueness problems.

math.CA↗

arXiv2025

Generic measures with slowly decaying Fourier coefficients

We investigate threshold phenomena in weighted $\ell^2$-spaces and characterize the critical regimes where elements with either small support or maximally bad range can be constructed. Our results are shown to be optimal in several respects, and our proofs principally rely on techniques involving sparse Fourier spectrum. We further show that these seemingly pathological constructions are actually generic from certain categorical perspectives.

math.FA↗

arXiv2025

A generic threshold phenomena in weighted $\ell^2$

We consider threshold phenomenons in the context of weighted $\ell^2$-spaces. Our main result is a summable Baire category version of K\"orner's topological Ivashev-Musatov Theorem, which is proved to be optimal from several aspects.

math.FA↗

arXiv2025

Summable analogous to the Ivashev-Musatov Theorems

We investigate summable analogues of the classical Ivashev-Musatov Theorem and threshold phenomenons alike. In the setting of weighted $\ell^1$ and Orlicz sequence spaces, we exhibit elements with critically pathological support and range, both topologically and in the sense of measure theory. Our results complement earlier works of J. P. Kahane, Y. Katznelson and T. W. K\"orner.

math.FA↗

arXiv2025

Revisiting cyclic elements in growth spaces

We revisit the problem of characterizing cyclic elements for the shift operator in a broad class of radial growth spaces of holomorphic functions on the unit disk, focusing on functions of finite Nevanlinna characteristic. We provide results in the range of Dini regular weights, and in the regime of logarithmic integral divergence. Our proofs are largely constructive, enabling us to simplify and extend a classical result by Korenblum and Roberts, and a recent Theorem due to El-Fallah, Kellay, and Seip.

math.CV↗

arXiv2024

Fourier coefficients of normalized Cauchy transforms

We consider a uniqueness problem concerning the Fourier coefficients of normalized Cauchy transforms. These problems inherently involve proving a simultaneous approximation phenomenon and establishing the existence of cyclic inner functions in certain sequence spaces. Our results have several applications in different directions. First, we offer a new non-probabilistic proof of a classic theorem by Kahane and Katzenelson on simultaneous approximation. Secondly, we demonstrate the absence of uniform admissible majorants of Fourier coefficients in de Branges-Rovnyak spaces.

math.CV↗

arXiv2024

Asymptotic polynomial approximation in the Bloch space

We investigate asymptotic polynomial approximation for a class of weighted Bloch functions in the unit disc. Our main result is a structural theorem on asymptotic polynomial approximation in the unit disc, in the flavor of the classical Plessner Theorem on asymptotic values of meromorphic functions. This provides the appropriate set up for studying metric and geometric properties of sets E on the unit circle for which the following simultaneous approximation phenomenon occurs: there exists analytic polynomials which converge uniformly to zero on E and to a non-zero function in the weighted Bloch norm. We offer a characterization completely within the realm of real-analysis, establish a connection to removable sets for analytic Sobolev functions in the complex plane, and provide several necessary conditions in terms of entropy, Hausdorff content and condenser capacity. Furthermore, we demonstrate two principal applications of our developments, which go in different directions. First, we shall deduce a rather subtle consequence in the theme of smooth approximation in de Branges-Rovnyak spaces. Secondly, we answer some questions that were raised almost a decade ago in the theory of Universal Taylor series.

math.CV↗

arXiv2023

Shift invariant subspaces in the Bloch space

We consider weak-star closed invariant subspaces of the shift operator in the classical Bloch space. We prove that any bounded analytic function decomposes into two factors, one which is cyclic and another one generating a proper shift invariant subspace, satisfying a permanence property, which in a certain way is opposite to cyclicity. Singular inner functions play the crucial role in this decomposition. We show in several different ways that the description of shift invariant subspaces generated by inner functions in the Bloch spaces deviates substantially from the corresponding description in the Bergman spaces, provided by the celebrated Korenblum and Roberts Theorem. Furthermore, the relationship between invertibility and cyclicity is also investigated and we provide an invertible function in the Bloch space which is not cyclic therein. Our results answer several open questions stated in the early nineties.

math.FA↗

arXiv2023

Shift invariant subspaces in growth spaces and sets of finite entropy

We investigate certain classes of shift invariant subspaces in growth spaces on the unit disc of the complex plane determined by a majorant $w$, which include the classical Korenblum growth spaces. Our main result provides a complete description of shift invariant subspaces generated by Nevanlinna class functions in growth spaces, where we show that they are of Beurling-type. In particular, our result generalizes the celebrated Korenblum-Roberts Theorem. It turns out that singular inner functions play the decisive role in our description, phrased in terms of certain $w$-entropy conditions on the carrier sets of the associated singular measures, which arise in connection to boundary zero sets for analytic functions in the unit disc having modulus of continuity not exceeding $w$ on the unit circle. Furthermore, this enables us to establish an intimate link between shift invariant subspace generated by inner functions and the containment of the above mentioned analytic function spaces in the corresponding model spaces.

math.CV↗

arXiv2021

Constructions of some families of smooth Cauchy transforms

For a given Beurling-Carleson subset $E$ of the unit circle $\mathbb{T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on $E$ such that their Cauchy transforms have smooth extensions from $\mathbb{D}$ to $\mathbb{T}$. The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several particular families of such Cauchy transforms with a few applications in operator and function theory in mind. In one application, we give a new proof of irreducibility of the shift operator on certain Hilbert spaces of functions. In another application, we establish a permanence principle for inner factors under convergence in certain topologies. The applications lead to a self-contained duality proof of the density of smooth functions in a very large class of de Branges-Rovnyak spaces. This extends the previously known approximation results.

math.FA↗

arXiv2021

Inner functions, invariant subspaces and cyclicity in $\mathcal{P}^t(μ)$-spaces

We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(μ)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $μ$ is a measure supported in the closed unit disk and $\mathcal{P}^t(μ)$ is the span of analytic polynomials in the usual Lebesgue space $L^t(μ)$. Our measures define a range of spaces somewhere in between the Hardy and the Bergman spaces, and our results are thus a mixture of results from these two theories. For a large class of measures $μ$ we characterize the cyclic inner functions, and exhibit some interesting properties of invariant subspaces generated by non-cyclic inner functions. Our study is motivated by a connection with the problem of smooth approximations in de Branges-Rovnyak spaces.

math.FA↗

arXiv2021

On the problem of smooth approximations in de Branges-Rovnyak spaces and connections to subnormal operators

For the class of de Branges-Rovnyak spaces $\mathcal{H}(b)$ of the unit disk $\mathbb{D}$ defined by extreme points $b$ of the unit ball of $H^\infty$, we study the problem of approximation of a general function in $\mathcal{H}(b)$ by a function with an extension to the unit circle $\mathbb{T}$ of some degree of smoothness, for instance satisfying Hölder estimates or being differentiable. We will exhibit connections between this question and the theory of subnormal operators and, in particular, we will tie the possibility of smooth approximations to properties of invariant subspaces of a certain subnormal operator. This leads us to several computable conditions on $b$ which are necessary for such approximations to be possible. For a large class of extreme points $b$ we use our result to obtain explicit necessary and sufficient conditions on the symbol $b$ which guarantee the density of functions with differentiable boundary values in the space $\mathcal{H}(b)$. These conditions include an interplay between the modulus of $b$ on $\mathbb{T}$ and the spectrum of its inner factor.

math.FA↗

arXiv2021

On model spaces and density of functions smooth on the boundary

We characterize the model spaces $K_Θ$ in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of $Θ$ is concentrated on a countable union of Beurling-Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to Beurling-Carleson sets, then even larger classes of functions, such as Hölder classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in $K_Θ$ and related spaces, the existence of a smooth approximant is obtained through a constructive method.

math.FA↗

arXiv2021

An abstract approach to approximations in spaces of pseudocontinuable functions

We give an abstract approach to approximations with a wide range of regularity classes $X$ in spaces of pseudocontinuable functions $K^p_\vartheta$, where $\vartheta$ is an inner function and $p>0$. More precisely, we demonstrate a general principle, attributed to A. B. Aleksandrov, which asserts that if a certain linear manifold $X$ is dense in the space of pseudocontinuable functions $K^{p_0}_\vartheta$, for some $p_0>0$, then $X$ is in fact dense in $K^p_{\vartheta}$, for all $p>0$. %This allows for generalizations of the recent result on density by functions with smooth boundary extensions. Moreover, for a rich class of Banach spaces of analytic functions $X$, we describe the precise mechanism that determines when $X$ is dense in a certain space of pseudocontinuable functions. As a consequence, we obtain an extension of Aleksandrov's density theorem to the class of analytic functions with uniformly convergent Taylor series.

math.FA↗

arXiv2021

Sparse Lerner operators in infinite dimensions

We use the principle of almost orthogonality to give a new and simple proof that a sparse Lerner operator is bounded on a matrix- or operator-weighted space $L_W^{2}(\mu)$, where $\mu$ is a doubling measure on $\R^d$ if and only if the weight $W$ satisfies the Muckenhoupt $A_2(\mu)$-condition, restricted to the sparse collection in question. Our method extends to the infinite-dimensional setting, thus allowing for applications to the multi-parameter setting. For the class of Muckenhoupt $A_2$-weights, we obtain bounds in terms of mixed $A_{2}(\mu)$-$A_{\infty}(\mu)$-conditions, which is independent of dimension and agrees with the best known bound in the finite-dimensional vectorial setting. As an application, we prove a matrix-weighted bound for the maximal Bergman projection, where we obtain a new sharper bound in terms of the B\'ekoll\'e-Bonami characteristic. Furthermore, we consider commutators of sparse Lerner operators on operator-valued weighted $L^{2}$-spaces and some applications to multi-parameters

math.FA↗

arXiv2020

Bloch functions and Bekollé-Bonami weights

We study analogues of well-known relationships between Muckenhoupt weights and $BMO$ in the setting of Bekollé-Bonami weights. For Bekollé-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekollé-Bonami weight, which in particular reveals an intimate connection between Bekollé-Bonami weights and Bloch functions. On the open problem of characterizing the closure of bounded analytic functions in the Bloch space, we provide a counter-example to a related recent conjecture. This shed light into the difficulty of preserving harmonicity in approximation problems in norms equivalent to the Bloch norm. Finally, we apply our results to study certain spectral properties of Cesaró operators.

math.CV↗

arXiv2019

Generalized Cesàro operators: geometry of spectra and quasi-nilpotency

For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk we prove that the spectrum of a generalized Cesàro operator $T_g$ is unchanged if the symbol $g$ is perturbed to $g+h$ by an analytic function $h$ inducing a quasi-nilpotent operator $T_h$, i.e. spectrum of $T_h$ equals $\{0\}$. We also show that any $T_g$ operator which can be approximated in the operator norm by an operator $T_h$ with bounded symbol $h$ is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function $g \in$ BMOA to be in the BMOA-norm closure of $H^{\infty}$. This condition turns out to be equivalent to quasi-nilpotency of the operator $T_g$ on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of $T_{g}$ operators.

math.FA↗