Efficiency of the weak Rescaled Pure Greedy Algorithm

2021 ◽  
pp. 2150001
Author(s):  
Bing Jiang ◽  
Peixin Ye
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Error Estimate ◽  
Banach Spaces ◽  
Greedy Algorithm ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Pure Greedy Algorithm ◽  
Sufficient Conditions For Convergence ◽  
Sufficient And Necessary Conditions

We study the efficiency of the Weak Rescaled Pure Greedy Algorithm (WRPGA) with respect to a dictionary in a Hilbert space or more generally a Banach space. We obtain the sufficient and necessary conditions for the convergence of WRPGA for any element and any dictionary. This condition is weaker than the sufficient conditions for convergence of the Weak Pure Greedy Algorithm (WPGA). In addition, we establish the noisy version of the error estimate for the WRPGA, which implies the results on sparse classes obtained in [G. Petrova, Rescaled pure greedy algorithm for Hlibert and Banach spaces, Appl. Comput. Harmon. Anal. 41(3) (2016) 852–866].

scholarly journals Bounded sets in the range of anX∗∗-valued measure with bounded variation

2000 ◽  
Vol 23 (1) ◽  
pp. 21-30 ◽  
Author(s):  
B. Marchena ◽  
C. Piñeiro
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Bounded Variation ◽  
Necessary Conditions ◽  
Bounded Set ◽  
Sufficient And Necessary Conditions ◽  
Bounded Sets

LetXbe a Banach space andA⊂Xan absolutely convex, closed, and bounded set. We give some sufficient and necessary conditions in order thatAlies in the range of a measure valued in the bidual spaceX∗∗and having bounded variation. Among other results, we prove thatX∗is a G. T.-space if and only ifAlies inside the range of someX∗∗-valued measure with bounded variation wheneverXAis isomorphic to a Hilbert space.

scholarly journals SPARSE APPROXIMATION AND RECOVERY BY GREEDY ALGORITHMS IN BANACH SPACES

2014 ◽  
Vol 2 ◽  
Author(s):  
V. N. TEMLYAKOV
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Greedy Algorithm ◽  
Matching Pursuit ◽  
Greedy Algorithms ◽  
Sparse Approximation ◽  
Trigonometric System ◽  
Space Setting ◽  
Redundant Dictionaries

AbstractWe study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty $.

scholarly journals On projection constant problems and the existence of metric projections in normed spaces

2001 ◽  
Vol 6 (7) ◽  
pp. 401-411 ◽  
Author(s):  
Entisarat El-Shobaky ◽  
Sahar Mohammed Ali ◽  
Wataru Takahashi
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Metric Projection ◽  
Infinite Matrix ◽  
Normed Spaces ◽  
Reflexive Banach Spaces ◽  
Linear Spaces ◽  
Sufficient And Necessary Conditions ◽  
Projection Constant

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spacesl p,1≤p<∞andc 0. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator froml p,1≤p<∞orc 0onto anyone of their maximal proper subspaces.

scholarly journals Asymptotic behaviour of non-autonomous dissipative systems in Hilbert space

1997 ◽  
Vol 62 (1) ◽  
pp. 93-103
Author(s):  
Song Guozhu
Keyword(s):  
Hilbert Space ◽  
Asymptotic Behaviour ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Integral Solution ◽  
Dissipative Systems ◽  
Dissipative Operators ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Sufficient And Necessary Conditions

AbstractIn this paper we discuss the asymptotic behaviour, as t → ∞, of the integral solution u(t) of the non-linear evolution equation where {A(t)}t≥0 is a family of m-dissipative operators in a Hilbert space H, and g ∈ Lloc (0, ∞ H).We give some sufficient conditions and some sufficient and necessary conditions to ensure that are weakly convergent.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

scholarly journals Right and Left Weyl Operator Matrices in a Banach Space Setting

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Aichun Liu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Operator Matrix ◽  
Weyl Operator ◽  
Operator Matrices ◽  
Space Setting ◽  
Hamiltonian Operators ◽  
Equivalent Conditions

Let X i , Y i i = 1,2 be Banach spaces. The operator matrix of the form M C = A C 0 B acting between X 1 ⊕ X 2 and Y 1 ⊕ Y 2 is investigated. By using row and column operators, equivalent conditions are obtained for M C to be left Weyl, right Weyl, and Weyl for some C ∈ ℬ X 2 , Y 1 , respectively. Based on these results, some sufficient conditions are also presented. As applications, some discussions on Hamiltonian operators are given in the context of Hilbert spaces.

Dynamics of an autonomous Gilpin–Ayala competition model with random perturbation

2021 ◽  
pp. 2050043
Author(s):  
Jing Fu ◽  
Qixing Han ◽  
Daqing Jiang ◽  
Yanyan Yang
Keyword(s):  
White Noise ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Necessary Conditions ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Interacting Species ◽  
Sufficient And Necessary Conditions ◽  
Global Positive Solution

This paper discusses the dynamics of a Gilpin–Ayala competition model of two interacting species perturbed by white noise. We obtain the existence of a unique global positive solution of the system and the solution is bounded in [Formula: see text]th moment. Then, we establish sufficient and necessary conditions for persistence and the existence of an ergodic stationary distribution of the model. We also establish sufficient conditions for extinction of the model. Moreover, numerical simulations are carried out for further support of present research.

scholarly journals Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 133
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Main Question ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Taylor Polynomials

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

scholarly journals Strongly Extreme Points and Middle Point Locally Uniformly Convex in Orlicz Spaces Equipped with s-Norm

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Yunan Cui ◽  
Yujia Zhan
Keyword(s):  
Banach Spaces ◽  
Orlicz Space ◽  
Necessary Conditions ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Uniformly Convex ◽  
Middle Point ◽  
Sufficient And Necessary Conditions

As is well known, the extreme points and strongly extreme points play important roles in Banach spaces. In this paper, the criterion for strongly extreme points in Orlicz spaces equipped with s-norm is given. We complete solved criterion-Orlicz space that generated by Orlicz function. And the sufficient and necessary conditions for middle point locally uniformly convex in Orlicz spaces equipped with s-norm are obtained.

scholarly journals A Distributed Approach to the Evasion Problem

Algorithms ◽  
2020 ◽  
Vol 13 (6) ◽  
pp. 149
Author(s):  
Denis Khryashchev ◽  
Jie Chu ◽  
Mikael Vejdemo-Johansson ◽  
Ping Ji
Keyword(s):  
Distributed Algorithm ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Movement Pattern ◽  
Sheaf Theory ◽  
Distributed Approach ◽  
Sufficient And Necessary Conditions ◽  
Evasion Problem ◽  
Over Time ◽  
Path Detection

The Evasion Problem is the question of whether—given a collection of sensors and a particular movement pattern over time—it is possible to stay undetected within the domain over the same stretch of time. It has been studied using topological techniques since 2006—with sufficient conditions for non-existence of an Evasion Path provided by de Silva and Ghrist; sufficient and necessary conditions with extended sensor capabilities provided by Adams and Carlsson; and sufficient and necessary conditions using sheaf theory by Krishnan and Ghrist. In this paper, we propose three algorithms for the Evasion Problem: one distributed algorithm extension of Adams’ approach for evasion path detection, and two different approaches to evasion path enumeration.

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