In this paper, we revisit the Riemann-Liouville analytic semigroup. In particular, we completely characterize the membership in the Schatten class S r ${\mathcal{S}}<^>{r}$ on L 2(0, 1), as well as the membership in the class of nuclear operators on L p (0, 1), p >= 1, and the membership in the ideal of absolutely r-summing operators for any r >= 1.

This is a contemporary treatment of composition operators on Banach spaces of analytic functions in one complex variable. It provides a step-by-step introduction, starting with a review (including full proofs) of the key tools needed, and building the theory with a focus on Hardy and Bergman spaces. Several proofs of operator boundedness (Littlewood's principle) are given, and the authors discuss approaches to compactness issues and essential norm estimates (Shapiro's theorem) using different tools such as Carleson measures and Nevanlinna counting functions. Membership of composition operators in various ideal classes (Schatten classes for instance) and their singular numbers are studied. This framework is extended to Hardy-Orlicz and Bergman-Orlicz spaces and finally, weighted Hardy spaces are introduced, with a full characterization of those weights for which all composition operators are bounded. This will be a valuable resource for researchers and graduate students working in functional analysis, operator theory, or complex analysis.

We propose VQ-$S_n$, a post-training neural network compression algorithm that combines vector quantization with element reordering. Sorting the elements of each weight vector in ascending order provably minimizes the pairwise distances between vectors via the Rearrangement Inequality, and empirically reduces $k$-means clustering error by $5.7$--$6.1\times$, yielding codebooks with substantially lower quantization error than standard vector quantization, at the additional storage cost of the permutation itself. On $100{,}000$ VGG19 $3\times3$ kernels, sorting reduces quantization error by 5.7--6.1$\times$ at every codebook size, an effect that is independent of the data distribution.We evaluate VQ-$S_n$ and its variants (NLE=0,1,2) against pruning, scalar quantization, and standard vector quantization on image classification (VGG on CIFAR100, with a CIFAR10 ablation), object detection (7 architectures on COCO), and semantic segmentation (10 architectures on Cityscapes). On CIFAR100, VQ-$S_n$ NLE=1 compresses VGG19 to $C_r = 0.100$ with $<5\%$ accuracy loss, achieving 40--44\% lower $C_r$ than VQ-Id across VGG11--16. On Cityscapes, VQ-$S_n$ NLE=1 achieves $C_r = 0.087$--$0.097$ on six of ten architectures, a consistent 1.9--2.2$\times$ improvement over scalar quantization, while VQ-Id never reaches usable quality on any of these six. Across all 21 model configurations, VQ-$S_n$ NLE=0 outperforms VQ-Id at every codebook size with zero exceptions.We characterize the inference overhead via hand-written CUDA decode kernels on an NVIDIA RTX~3060 across 7~models (VGG and ResNet families, 111~convolutional layers): centroid lookup and permutation unsort cost 0.3~ns and 0.1~ns per kernel vector respectively, yielding $<$1\% per-layer overhead for deep $512\to512$ layers. Full-model VQ-$S_n$ overhead ranges from 7.4\% (ResNet-50, 44\% 3$\times$3 parameters) to 35.1\% (ResNet-34, 97\% 3$\times$3), with all models above 150~FPS at 224$\times$224.

The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit disc. In this sense, Bergman spaces were introduced several years ago contributing to broaden the picture of this theory. In the paper presenting these new family of spaces, some of its most essential questions were considered. Among them, some partial estimates of the norm of the pointwise evaluation functional were given. In this work, we introduce a version of the Riemann-Liouville semigroup acting on these spaces, and, with this new tool, we are able to estimate the norm of this functional.

We give a complete characterization of the sequences beta = (beta n) of positive numbers for which all composition operators on H2(beta) are bounded, where H2(beta) is the space of analytic functions f on the unit disk (R) such that Sigma infinity n=0 |an|2 beta n < +infinity if f (z) = Sigma infinity n=0anzn. We prove that all composition operators are bounded on H2(beta) if and only if beta is essentially decreasing and slowly oscillating. We also prove that every automorphism of the unit disk induces a bounded composition operator on H2(beta) if and only if beta is slowly oscillating. We give applications of our results.

Since their introduction in 1997, Hardy spaces of Dirichlet series have been broadly studied. The increasing interest which they sparked motivated the introduction of new such spaces, as the Bergman spaces A mu considered here, with mu a probability measure. Similarly, recent lines of research have focused on the study of some classical operators acting on these spaces, like the Volterra operator V. In this work, we introduce a new family of spaces of Dirichlet series, the Bloch mu -spaces. We can provide, in terms of those spaces, a sufficient condition for this Volterra operator to act boundedly on the spaces A mu. We also establish a necessary condition for a specific choice of mu. Sufficient and necessary conditions for compactness are also proven. The non-membership in Schatten classes is established, as well as a radicality result for some Bloch space. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

In this paper, we focus on Carleson embeddings from Bergman spaces Ap into Lp(μ), where μ is a positive measure on the unit disk. We describe when this injection is r-summing on Ap. We complete the full characterization of such operators when p>1, and r≥1. As an immediate application, we get the characterization of absolutely summing weighted composition operators on Bergman spaces. In passing we also prove a new connection between the boundedness of the Berezin transform and the Carleson embedding on Bergman spaces.

We characterize the integration operators V_g with symbol g for which V_g acts as an absolutely summing operator on weighted Bloch spaces ℬ^β and on weighted Bergman spaces 𝒜^p_α . We show that V_g is r -summing on 𝒜^p_α , 1 ≤ p <∞ , if and only if g belongs to a suitable Besov space. We also show that there is no non trivial nuclear Volterra operators V_g on Bloch spaces and on Bergman spaces.

We give a short proof of a weighted version of the discrete Hardy inequality. This includes the known case of classical monomial weights with optimal constant. The proof is based on the ideas of the short direct proof given recently in [P. Lefèvre, Arch. Math. (Basel) , 114 , No. 2, 195–198 (2020)].

We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $\beta$ every symbol $\varphi \colon \mathbb{D} \to \mathbb{D}$ with $\varphi \in H^2 (\beta)$ induces a bounded composition operator $C_\phi$ on the weighted Hardy space $H^2 (\beta)$. We give partial answers and investigate when $H^2 (\beta)$ is an algebra. We answer negatively to another question in showing that there are a sequence $\beta$ and $\varphi \in H^2 (\beta)$ such that $\| \varphi \|_\infty < 1$ and the composition operator $C_\varphi$ is not bounded on $H^2 (\beta)$. In a second part, we show that for $p \neq 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space $\ell^p_A$, and even on the weighted versions of this space.

We characterize the symbols φ for which there exists a weight w such that the weighted composition operator MwCφ is compact on the weighted Bergman space Bα2. We also characterize the symbols for which there exists a weight w such that MwCφ is bounded but not compact. We also investigate when there exists w such that MwCφ is Hilbert-Schmidt on Bα2.

We characterize p-summing composition operators from a Bloch space B-mu to another such space B-beta, where mu, beta > 0. The corresponding result on little Bloch-type spaces is also proved. We construct an example of a conformal mapping from D into itself which has a contact point with the unit circle T, and induces a compact composition operator, that fails to be p-summing for any p >= 1. We also detail the case of lens maps. Moreover we explore the case of weighted composition operators and give characterizations for a class of weights. We also show that compactness of a composition operator on B-beta and B-0(beta) implies its compactness on Bergman spaces.

In this paper, we propose an image tampering localization algorithm using semi-fragile watermarking and Error-Locating codes in the DWT domain. By introducing different families of codes, we show the benefit in terms of image tampering localization and complexity of using control code error localization as an authentication function. Indeed, we first experimentally show that error localization block codes is as precise as using classical error correcting codes (Reed-Solomon and BCH codes) to locate image tampering. However, their corresponding decoding algorithms complexity is at least quadratic which make them impractical for some real time applications. To solve this problem, we introduce error-control codes called Error-Locating codes where error localization is reduced to a single syndrome computation performed with low number of binary operations (detailed later in the paper). We provide comparisons of image quality and tampering localization performances using error-detection, error-localization and error correction approaches with different error control codes. (c) 2021 Elsevier B.V. All rights reserved.

This chapter analyses the principle of digital watermarking of images through the prism of the use of error-correcting codes in a very specific framework, namely the so-called robust watermark. It introduces a more original code and shows how a specific code can respond to a particular problem, the problem of cropping. Over the years, several watermarking paradigms have emerged due to the protection requirements of many applications. The chapter concentrates on so-called robust watermarking, which constitutes the classical application framework for the use of codes in watermarking. It discusses a simple use case based on watermarking by index modulation applied to color images. Error-correcting codes are powerful tools in information theory. The aim of robust watermarking is to optimize three properties in particular: the maximum amount of information that an image can contain, the invisibility of the mark and the robustness of the mark to image changes and, in some cases, security.

Ce chapitre présente les différentes stratégies permettant d'assurer l'invisibilité d'un message enfoui dans une image couleur. Le terme invisibilité concerne ici l'aspect visuel. Nous présentons ici les différentes solutions qui ont été développées tentant de prendre en compte l’aspect la modélisation du système visuel humain (SVH), afin de limiter les dégradations pour les images couleur. Nous verrons que le défi se traduit par le bon choix de la direction couleur d'insertion, même lorsque la méthode repose sur une représentation dédiée à la couleur (comme avec les Quaternions).

We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk $\D$. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the ``cusp map'' and the lens maps, acting on weighted Dirichlet spaces.

Ce chapitre présente le principe du tatouage numérique des images à travers le prisme de l'utilisation des codes correcteurs dans le cadre du tatouage dit robuste. Durant ce chapitre, les codes correcteurs de Hamming classiques ainsi que les codes BCH et les codes de Reed-Solomon sont présentés associés aux structures d'erreurs aléatoires et par paquet. Afin d’illustrer l'impact et l'intérêt de l’utilisation des codes correcteurs, nous déployons un cas d'usage simple basé sur un tatouage par modulation d'index. Nous discutons notamment des différences de comportement concernant la robustesse selon l'attaque subie et le code utilisé.

We study when multiplication by a weight can turn a non-compact composition operator on H 2 into a compact operator, and when it can be in Schatten classes. The q-summing case in H p is considered. We also study when this multiplication can turn a compact composition operator into a non-compact one. MSC 2010 primary: 47B33 ; secondary: 46B28

We characterize the (essentially) decreasing sequences of positive numbers $\beta$ = ($\beta$ n) for which all composition operators on H 2 ($\beta$) are bounded, where H 2 ($\beta$) is the space of analytic functions f in the unit disk such that $\infty$ n=0 |c n | 2 $\beta$ n < $\infty$ if f (z) = $\infty$ n=0 c n z n. We also give conditions for the boundedness when $\beta$ is not assumed essentially decreasing.

We characterize the nuclear composition operators C(phi)f = f omicron phi on the classical and little Bloch spaces. In addition, we construct an example of a conformal mapping of the unit disk D into itself which has a contact point with the unit circle T, and induces a nuclear composition operator.