Unconditional Basis Research Articles

The uniqueness of unconditional basis of the 2-convexified Tsirelson space, revisited

One of the hallmarks in the study of the classification of Banach spaces with a unique (normalized) unconditional basis was the unexpected result by Bourgain, Casazza, Lindenstrauss, and Tzafriri from their 1985 Memoir that the 2-convexified Tsirelson space T(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^{(2)}$$\\end{document} had that property (up to equivalence and permutation). Indeed, on one hand, finding a “pathological” space (i.e., not built out as a direct sum of the only three classical sequence spaces with a unique unconditional basis) shattered the hopeful optimism of attaining a satisfactory description of all Banach spaces which enjoy that important structural feature. On the other hand it encouraged furthering a research topic that had received relatively little attention until then. After forty years, the advances on the subject have shed light onto the underlying patterns shared by those spaces with a unique unconditional bases belonging to the same class, which has led to reproving the original theorems with fewer technicalities. Our motivation in this note is to revisit the aforementioned result on the uniqueness of unconditional basis of T(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^{(2)}$$\\end{document} from the current state-of-art of the subject and to fill in some details that we missed from the original proof.

2-Rotund norms for unconditional and symmetric sequence spaces

A reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2R norm and embeds into a reflexive space with a 1-symmetric 2R norm. Partial results on 1-symmetric 2R renormings of spaces with a symmetric basis are obtained.

Solid bases and functorial constructions for (p-)Banach spaces of analytic functions

Abstract Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials $\{z^n\}_{n\geq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of $\mathcal{X}^\mathrm{max}$ (and $\mathcal{X}^\mathrm{min}$ ), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.

ფინანსებზე ხელმისაწვდომობის სახელმწიფო მხარდაჭერა საქართველოში, გამოწვევები და შესაძლებლობები

Small and medium-sized businesses are the most important lever for the country's economic development. It is the country's main source of employment and income for economic development and public welfare. Therefore, encouraging and developing entrepreneurship is a top priority for any country. The introduction of continuous innovation forms the unconditional basis for enterprises' long-term and sustainable success. Innovation needs technological perfection and renewal, which requires financial resources. Businesses often do not have the internal resources to introduce innovations and increase production. Providing enterprises with access to financial resources is a priority direction for the country. The paper aims to study the existing mechanisms of state financial support for entrepreneurship in Georgia, determine their effectiveness, identify potential opportunities, and develop recommendations for improving access to financial resources. During the research process, articles by both foreign and Georgian scientists, published in local and international scientific publications, government reports and studies of international institutions, authoritative organizations, and statistical data were used. The methods of analysis and synthesis, including quantitative and qualitative analysis, are used as research methods. The paper identifies the main problems that appear as barriers for Georgian entrepreneurs in receiving financial support and elaborates on specific recommendations to solve them. The paper emphasizes the fact that, at the current stage in Georgia, despite the reforms implemented by the state, finding a source of financial resources remains a problem. When developing financial support programs, more attention should be paid and the needs of entrepreneurs should be investigated in depth. This will help increase the efficiency of grant projects, state programs, and business development. Proper access to finance will improve the competitiveness of small and medium-sized businesses and will contribute to the country's economic development.

The public danger of an act in an intersectoral dimension

Relevance. In the criminal law doctrine, the sign of public danger is defined as fundamental, allowing one to distinguish a crime from other offenses. The increasing complexity of existing legal relations and the unsystematic development of sectoral rule-making call into question this dogmatic judgment. The feasibility of using this characteristic as an unconditional basis for criminalization requires verification.The purpose of the study is to assess the prospects for using the sign of public danger to criminalize an act and distinguish a crime from other offenses.Objectives: study the genesis of the material sign of a crime with the identification of its criminal-political function at the stage of its emergence and in the process of evolution.Methodology. The methodological basis of the study was the universal dialectical method of cognition of phenomena and processes of the surrounding reality. In the course of developing the theoretical provisions of the work, historical-legal, formal-logical, formal-legal, semantic and other methods were also used.Results. Initially, the sign of public danger had a fundamentally different criminal and political meaning. It opened up wide possibilities for applying the law by analogy. The question of distinguishing the criminal from the noncriminal through its use was not raised. Subsequently, the demarcation of protective legislation and attempts to give sectoral uniqueness to criminal law by using a material sign of the social danger of an act did not resolve the issue of the boundaries of criminalization and the limits of the state’s punitive power; on the contrary, the situation worsened. Using examples from practice, based on the judgments of researchers about the essence of the social danger of an act, the authors of the article question the advisability of using this feature as an unconditional basis for the criminalization of an act.Conclusions. Based on the results of the study, the authors come to the conclusion that in the process of criminalization of an act, the focus of attention should be shifted from an idealized sign of social danger to the principles of criminalization: economy of repression, formal certainty, intersectoral consistency and proportionality, as well as the expediency of criminalization (declaring a crime) from the perspective of the possibilities of criminal the right to influence real relationships, transforming them qualitatively.

Linear versus nonlinear forms of partial unconditionality of bases

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory.

Delta‐points and their implications for the geometry of Banach spaces

AbstractWe show that the Lipschitz‐free space with the Radon–Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to . Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of , with a ‐point. Building on these two results, we are able to renorm every infinite‐dimensional Banach space to have a ‐point. Next, we establish powerful relations between existence of ‐points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of ‐points for the asymptotic geometry of Banach spaces. In addition, we prove that if is a Banach space with a shrinking ‐unconditional basis with , or if is a Hahn–Banach smooth space with a dual satisfying the Kadets–Klee property, then and its dual fail to contain ‐points. In particular, we get that no Lipschitz‐free space with a Hahn–Banach smooth predual contains ‐points. Finally, we present a purely metric characterization of the molecules in Lipschitz‐free spaces that are ‐points, and we solve an open problem about representation of finitely supported ‐points in Lipschitz‐free spaces.

Несоблюдение правил подсудности гражданских дел: процессуально-правовые последствия

This article analyzes the procedural and legal consequences of non-compliance with the rules of jurisdiction in civil proceedings at its various stages. The author of the work concludes that consideration of the issue of jurisdiction of the application is the prerogative of the court of first instance. It is noted that violation of the rules of jurisdiction cannot be an unconditional basis for the cancellation of a judicial act. A doctrinal definition of the concept of “violation of the rules of jurisdiction” and a classification of violations are proposed.

Unconditional bases and Daugavet renormings

We prove that the space ℓ2 – and more generally every infinite dimensional Banach space with an unconditional Schauder basis – can be renormed to have a Daugavet point. As a starting point, we introduce a new diametral notion for points of the unit sphere of Banach spaces, that complements the notion of Δ-points, but is weaker than the notion of Daugavet points. We show that this notion can be used to provide another geometric characterization of the Daugavet property, as well as to recover – and even to provide new – results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Then, we observe that the study of this notion naturally leads to new renorming ideas in the context, and combine these with previous techniques from the literature to show that Daugavet points may exist even in super-reflexive spaces.

On the Properties of the Root Vector Function Systems of a th-Order Dirac Type Operator with an Integrable Potential

We consider a Dirac type operator with matrix coefficients. Estimates for the root vector functions are established, and criteria for the Bessel property and the unconditional basis property of the root vector function systems of this operator in the space L2m2(G) , where G=(a,b)⊂R is a finite interval, are obtained.

On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function px, which belongs to the space L20,1. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when px≡0) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions px, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L20,1, is dense within the space L20,1. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.

On a sufficient condition for the existence of unconditional bases of reproducing kernels in Fock type spaces with nonradial weights

On a sufficient condition for the existence of unconditional bases of reproducing kernels in Fock type spaces with nonradial weights

Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on Rd. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.In the second part of the paper we study nonlinear m-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for d≥2, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for m-term approximation can still be derived.

On the exponential stability of Beck's Problem on a star-shaped graph

We deal with the as yet unresolved exponential stability problem for Beck's Problem on a metric star graph with three identical edges. The edges are stretched Euler–Bernoulli beams which are simply supported with respect to the outer vertices. At the inner vertex we have viscoelastic damping acting on the slopes of the edges. We carry out a complete spectral analysis of the system operator associated with the abstract spectral problem in Hilbert space. Within this framework it is shown that the eigenvectors have the property of forming a Riesz (i.e. an unconditional) basis, which makes it possible to directly deduce the exponential stability of the corresponding C0-semigroup using spectral information for the system operator alone. A physically interesting conclusion is that the particular choice of vertex conditions ensures the exponential stability even when the elasticity acting on the slopes of the edges is absent.

Application of the Principle of Legal Certainty to Legal Relations in the field of Customs by the Court of Justice of the European Union and the Court of the Eurasian Economic Union

The principle of legal certainty, according to which the established rules should allow interested parties to clearly understand the scope of the obligations imposed on them, has found wide application in the practice of the Court of Justice of the European Union (CJEU) and the Court of the Eurasian Economic Union (EAEU Court) on customs legal relations. The CJEU and the EAEU Court not only use this principle as an additional method of argumentation, but also as an independent basis for monitoring the legality of secondary law acts adopted by the bodies of these integration associations. As the analysis of the practice of the CJEU shows, this principle has two main aspects: material, expressed in the requirement of clarity of the act, and temporal, associated primarily with the prohibition of retroactive effect of the EU act. At the same time, within the framework of the material aspect, the EU Court draws attention not only to the absence of internal contradictions in the act, but also to compliance with the requirement to publish the act and to indicate the legal framework. In the practice of the EAEU Court (and earlier the EurAsEC Court) on customs legal relations, compliance with the principle of legal certainty is checked in terms of clarity and consistency of the Commission’s decisions. As the practice of the EAEU Court devoted to the classification of certain types of goods shows, a violation of these requirements may be a consequence of both the confusion in the content of one classification decision of classification features characteristic of different commodity items, and the absence in the classification decision of the necessary classification features that allow distinguishing the goods subject to classification in different commodity items. In these cases, non-compliance with the requirement of legal certainty serves as an unconditional basis for recognizing the Commission’s decisions as inconsistent with the law of the Union.

Power dilation systems {𝑓(𝑧^{𝑘})}_{𝑘∈ℕ} in Dirichlet-type spaces

Power dilation systems { f ( z k ) } k ∈ N \{f(z^k)\}_{k\in \mathbb {N}} in Dirichlet-type spaces D t ( t ∈ R ) \mathcal {D}_t\ (t\in \mathbb {R}) are treated. When t ≠ 0 t\neq 0 , it is proved that a system of functions { f ( z k ) } k ∈ N \{f(z^k)\}_{k\in \mathbb {N}} is orthogonal in D t \mathcal {D}_t only if f = c z N f=cz^N for some constant c c and some positive integer N N . Complete characterizations are also given of unconditional bases and frames formed by power dilation systems of Dirichlet-type spaces. Finally, these results are applied to the operator theoretic case of the moment problem on Dirichlet-type spaces.

Haar Frame Characterizations of Besov–Sobolev Spaces and Optimal Embeddings into Their Dyadic Counterparts

We study the behavior of Haar coefficients in Besov and Triebel–Lizorkin spaces on {mathbb R}, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness s<1, in which the spaces F^s_{p,q} and B^s_{p,q} are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that 1/p<s<1 and fin B^s_{p,q}, we actually prove that the usual Haar coefficient norm, Vert {2^jlangle f, h_{j,mu }rangle }_{j,mu }Vert _{b^s_{p,q}} remains equivalent to Vert fVert _{B^s_{p,q}}, i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case s=1 and q=infty , we show that such an expression gives an equivalent norm for the Sobolev space W^{1}_p({mathbb R}), 1<p<infty , which is related to a classical result by Bočkarev. Finally, in several endpoint cases we give optimal inclusions between B^s_{p,q}, F^s_{p,q}, and their dyadic counterparts.

Band-diagonal operators on Banach lattices: Matrix dynamics and invariant subspaces

We address the existence of non-trivial closed invariant ideals for positive operators defined on Banach lattices whose order is induced by an unconditional basis. In particular, for band-diagonal positive operators such existence is characterized whenever their matrix representations meet a positiveness criteria. For more general classes of positive operators, sufficient conditions are derived proving, particularly, the sharpness of such results from the standpoint of view of the matrix representations. The whole approach is based on studying the behavior of the dynamics of infinite matrices and the localization of the non-zero entries. Finally, we generalize a theorem of Grivaux regarding the existence of non-trivial closed invariant subspaces for positive tridiagonal operators to a more general class of band-diagonal operators showing, in particular, that a large subclass of them have non-trivial closed invariant subspaces but lack non-trivial closed invariant ideals.

ON PROPERTIES OF SPACES WITH MARTINGALE MIXED NORM WITH SUMMABILITY EXPONENTS CLOSE TO ONE

For spaces of random variables with martingale mixed norm the theorem of A. Pelczynski on the absence of an unconditional basis in the space $$L_1(0,1)$$ is generalized and (in terms of summability exponents) the cases are studied when these spaces coincide with the space $$L_1$$ . 2020 Mathematics subject classification: Primary 60G46; Secondary 60G42, 46E30.

Existence of unconditional wavelet bases for Lp-norm over a local fields of positive characteristic

In this paper, we defined Calder[Formula: see text]n–Zygmund operator [Formula: see text] as [Formula: see text] on local fields ([Formula: see text]) under certain conditions on [Formula: see text] and then find the boundedness of operator [Formula: see text]. Using result on boundedness of [Formula: see text] and orthonormal wavelet basis of [Formula: see text], we provide unconditional wavelet bases for [Formula: see text]-norm over a local fields of positive characteristic by the wavelet coefficients.