<abstract xmlns="http://www.w3.org/1999/xhtml">

This note is dedicated to recalling the virtues and the important contributions in mathematics of mohamed zarabi who passed a way on mid december 2021.
</abstract>

Mohamed Zarrabi 1964-2021

<abstract xmlns="http://www.w3.org/1999/xhtml">

We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space <italic>H</italic>2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.
</abstract>

Laboratoire ADEF, Campus Universitaire de Saint-Jérôme

<italic>H</italic>∞ interpolation constrained by Beurling–Sobolev norms

<abstract xmlns="http://www.w3.org/1999/xhtml">

We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.
</abstract>

Carleson’s formula for some weighted Dirichlet spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

There exist two real valued periodic functions on the real line such that, for every <italic>x</italic> ∈ ℝ, <italic>f</italic>1(<italic>x</italic>) + <italic>f</italic>2(<italic>x</italic>) = <italic>x</italic>, but it is impossible to find two real valued periodic functions on the real line such that, for every <italic>x</italic> ∈ ℝ, <italic>f</italic>1(<italic>x</italic>) + <italic>f</italic>2(<italic>x</italic>) = <italic>x</italic>2. The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.
</abstract>

Sums and products of periodic functions

<abstract xmlns="http://www.w3.org/1999/xhtml">

Zarrabi proved in 1993 that if the spectrum of a contraction <italic>T</italic> on a Banach space is a countable subset of the unit circle 𝕋, and if 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mo>lim</mo></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></msub><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mrow><mrow><mo>‖</mo> <mrow><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow> <mo>‖</mo></mrow></mrow><mo>)</mo></mrow></mrow><mrow><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math>
<tex-math>{\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0</tex-math>
</alternatives>
</inline-formula>, then <italic>T</italic> is an isometry, so that ‖<italic>Tn</italic>‖ = 1 for every <italic>n</italic> ∈ ℤ. It is also known that if <italic>C</italic> is the usual triadic Cantor set then every contraction <italic>T</italic> on a Banach space such that <italic>Spec</italic>(<italic>T</italic> ) ⊂ 𝒞 satisfying 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>lim</mo><mi> </mi><mi>s</mi><mi>u</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></msub><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mrow><mrow><mo>‖</mo> <mrow><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow> <mo>‖</mo></mrow></mrow><mo>)</mo></mrow></mrow><mrow><msup><mrow><mi>n</mi></mrow><mi>α</mi></msup></mrow></mfrac><mo>&lt;</mo><mo>+</mo><mo>∞</mo></mrow></math>
<tex-math>\lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} &lt; + \infty</tex-math>
</alternatives>
</inline-formula>
 for some 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>&lt;</mo><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>−</mo><mo>log</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi> </mi><mo>log</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mo>−</mo><mo>log</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mrow></math>
<tex-math>\alpha &lt; {{\log \left( 3 \right) - \log \left( 2 \right)} \over {2\,\log \left( 3 \right) - \log \left( 2 \right)}}</tex-math>
</alternatives>
</inline-formula> is an isometry.
In the other direction an easy refinement of known results shows that if a closed <italic>E</italic> ⊂ 𝕋 is not a “strong <italic>AA</italic>+-set” then for every sequence (<italic>un)n</italic>≥1 of positive real numbers such that lim inf<italic>n</italic>→+∞<italic>un</italic> = + ∞ there exists a contraction <italic>T</italic> on some Banach space such that <italic>Spec</italic>(<italic>T</italic> )⊂ <italic>E</italic>, ‖<italic>T</italic>−<italic>n</italic>‖ = <italic>O</italic>(<italic>u</italic>n) as <italic>n</italic> → + ∞ and supn≥1 ‖<italic>T</italic>−n‖ = + ∞.
We show conversely that if <italic>E</italic> ⊂ 𝕋 is a strong <italic>AA</italic>+-set then there exists a nondecreasing unbounded sequence (<italic>u</italic>n)n≥1 such that for every contraction <italic>T</italic> on a Banach space satsfying <italic>Spec</italic>(<italic>T</italic>) ⊂ <italic>E</italic> and ‖<italic>T</italic>−n ‖ = <italic>O</italic>(<italic>u</italic>n) as <italic>n</italic> → + ∞ we have <italic>supn</italic>>0 ‖<italic>T</italic>−n ‖ ≤ <italic>K</italic>, where <italic>K &lt;</italic> + ∞ denotes the “<italic>AA</italic>+-constant” of <italic>E</italic> (closed countanble subsets of 𝕋 and the triadic Cantor set are strong <italic>AA</italic>+-sets of constant 1).
</abstract>

Negative Powers of Contractions Having a Strong <italic>AA</italic>+ Spectrum

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we study the cyclicity problem with respect to the forward shift operator <italic>Sb</italic> acting on the de Branges–Rovnyak space ℋ (<italic>b</italic>) associated to a function <italic>b</italic> in the closed unit ball of <italic>H</italic>∞ and satisfying log(1− |<italic>b</italic>| ∈ <italic>L</italic>1(𝕋). We present a characterisation of cyclic vectors for <italic>Sb</italic> when <italic>b</italic> is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of <italic>Sb</italic> in this case, but we provide here an elementary proof. We also study the situation where <italic>b</italic> has the form <italic>b</italic> = (1+ <italic>I</italic>)/2, where <italic>I</italic> is a non-constant inner function such that the associated model space <italic>K</italic>I = ℋ (<italic>I</italic>) has an orthonormal basis of reproducing kernels.
</abstract>

Cyclicity in de Branges–Rovnyak spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice <italic>X</italic>, including weighted <italic>ℓ</italic>p spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔻</mi></mrow><mi>X</mi><mo>∞</mo></msubsup></mrow></math>
<tex-math>\mathbb{D}_X^\infty</tex-math>
</alternatives>
</inline-formula>
. Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔻</mi></mrow><mi>X</mi><mo>∞</mo></msubsup><mo>,</mo><mo>|</mo><msup><mrow><mi>ℂ</mi></mrow><mi>N</mi></msup><mo>=</mo><msup><mrow><mi>𝔻</mi></mrow><mi>N</mi></msup></mrow></math>
<tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N}</tex-math>
</alternatives>
</inline-formula>
 and the <italic>ℓ</italic>p-unit balls 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔻</mi></mrow><mi>X</mi><mo>∞</mo></msubsup><mo>,</mo><mo>|</mo><msup><mrow><mi>ℂ</mi></mrow><mi>N</mi></msup><mo>=</mo><msubsup><mrow><mi>𝔹</mi></mrow><mi>p</mi><mi>N</mi></msubsup></mrow></math>
<tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N</tex-math>
</alternatives>
</inline-formula>
, in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_004.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>=</mo><msup><mrow><mi>ℓ</mi></mrow><mi>p</mi></msup><mrow><mo>(</mo><mrow><msubsup><mrow><mi>ℤ</mi></mrow><mo>+</mo><mi>N</mi></msubsup><mo>,</mo><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>s</mi></msup></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right))</tex-math>
</alternatives>
</inline-formula>, <italic>s</italic> = (<italic>s</italic>1, <italic>s</italic>2, … ) and <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_005.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>=</mo><msup><mrow><mi>ℓ</mi></mrow><mi>p</mi></msup><mrow><mo>(</mo><mrow><msubsup><mrow><mi>ℤ</mi></mrow><mo>+</mo><mi>N</mi></msubsup><mo>,</mo><mi> </mi><mi> </mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>α</mi><mo>!</mo></mrow><mrow><mrow><mo>|</mo> <mi>α</mi> <mo>|</mo></mrow><mo>!</mo></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mi>t</mi></msup><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mo>|</mo> <mi>α</mi> <mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>s</mi></msup></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right)</tex-math>
</alternatives>
</inline-formula>, <italic>s,t</italic> ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.
</abstract>

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.
</abstract>

Volterra operator norms : a brief survey

<abstract xmlns="http://www.w3.org/1999/xhtml">

For a Bloch function <italic>f</italic> in the unit ball in ℂ<italic>n</italic>, we study the maximal locus of the Bloch norm of <italic>f</italic>; namely, the set <italic>Lf</italic> where the Bergman length of the gradient vector field of <italic>f</italic> attains its maximum. We prove that for <italic>n</italic> ≥<italic>,</italic> the set <italic>Lf</italic> consists of a finite union of real analytic sets with dimensions at most 2<italic>n −</italic> 2. This is not the case for <italic>n</italic> = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set <italic>Lf</italic>. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.
</abstract>

The Maximum Locus of the Bloch Norm

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Moroccan Journal of Pure and Applied Analysis

 The Moroccan Journal of Pure and Applied Analysis "MJPAA" (Print ISSN: 2605-6364) is a peer-reviewed open access journal published by the "Université Sidi Mohamed Ben Abdellah" (Sidi Mohamed Ben Abdallah University), covering all main-stream branches of pure and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, numerical analysis, optimization theory, and control theory.   MJPAA is published by Sidi Mohamed Ben Abdallah University in three issues in January, May and September.   Why subscribe and read    The journal covers a wide range of branches of mathematical analysis.   Pure and applied analysis topics are intensely studied by researchers throughout the world especially in Africa, the Arab and European countries.   An edition committee of high-level and international renown.   Editorial board representing different regions of the world and various research teams.      Why submit    No publication charges for the authors or their institutions.   The journal is available to be indexed in the most important databases.   The journal is among the best in its field in Africa and the Arab world.      Rejection Rate    45%      Archiving   Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.   Plagiarism Policy   The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process.   Publisher:Sidi Mohamed Ben Abdallah University (Route Imouzzer, BP 2626 Fez 30000, Morocco, Tel. : 0212 5 35 60 96 60/ 61)   Contact:Professor Youssef Akdim, Sidi Mohamed Ben Abdellah Universitymjpaa.edition1@usmba.ac.ma 

Moroccan Journal of Pure and Applied Analysis

Volume 9 (2023): Issue 2 (May 2023)

Mohamed Zarrabi 1964-2021

H^∞ interpolation constrained by Beurling–Sobolev norms

Carleson’s formula for some weighted Dirichlet spaces

Sums and products of periodic functions

Negative Powers of Contractions Having a Strong AA⁺ Spectrum

Cyclicity in de Branges–Rovnyak spaces

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

Volterra operator norms : a brief survey

The Maximum Locus of the Bloch Norm

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