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 Numerical ranges of conjugations and antilinear operators on a Banach space

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2715-2720
Author(s):  
Muneo Chō ◽  
Injo Hur ◽  
Ji Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unit Circle ◽  
Banach Spaces ◽  
Unit Disc ◽  
Numerical Range ◽  
Space Setting ◽  
Path Connectedness ◽  
The Given

In this paper, we prove that the numerical range of a conjugation on Banach spaces, using the connected property, is either the unit circle or the unit disc depending the dimension of the given Banach space. When a Banach space is reflexive, we have the same result for the numerical range of a conjugation by applying path-connectedness which is applicable to the Hilbert space setting. In addition, we show that the numerical ranges of antilinear operators on Banach spaces are contained in annuli.

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 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 TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

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.

scholarly journals Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2066
Author(s):  
Messaoud Bounkhel ◽  
Mostafa Bachar
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Space Setting ◽  
Nonlinear Anal ◽  
Finite Dimensional ◽  
In Trans ◽  
First Time ◽  
The Relationship ◽  
Primal Lower Nice Functions

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

scholarly journals m-ISOMETRIC OPERATORS ON BANACH SPACES

2010 ◽  
Vol 03 (01) ◽  
pp. 1-19 ◽  
Author(s):  
Ould Ahmed Mahmoud Sid Ahmed
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Natural Generalization ◽  
Basic Properties

We introduce the class of m-isometric operators on Banach spaces. This generalizes to Banach space the m-isometric operators on Hilbert space introduced by Agler and Stankus. We establish some basic properties and we introduce the notion of m-invertibility as a natural generalization of the invertibility on Banach spaces.

Bernstein and Markov-type inequalities for polynomials on real Banach spaces

2002 ◽  
Vol 133 (3) ◽  
pp. 515-530 ◽  
Author(s):  
GUSTAVO A. MUÑOZ ◽  
YANNIS SARANTOPOULOS
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Real Hilbert Space ◽  
Markov Type ◽  
Markov's Inequality ◽  
Markov’S Inequality ◽  
Derivatives Of ◽  
Derivative Of A Polynomial

In this work we generalize Markov's inequality for any derivative of a polynomial on a real Hilbert space and provide estimates for the second and third derivatives of a polynomial on a real Banach space. Our result on a real Hilbert space answers a question raised by L. A. Harris in his commentary on problem 74 in the Scottish Book [20]. We also provide generalizations of previously obtained inequalities of the Bernstein and Markov-type for polynomials with curved majorants on a real Hilbert space.

scholarly journals One-Bit Compressed Sensing by Greedy Algorithms

2016 ◽  
Vol 9 (2) ◽  
pp. 169-184 ◽  
Author(s):  
Wenhui Liu ◽  
Da Gong ◽  
Zhiqiang Xu
Keyword(s):  
Compressed Sensing ◽  
Greedy Algorithm ◽  
Numerical Experiments ◽  
Main Part ◽  
Matching Pursuit ◽  
Greedy Algorithms ◽  
Orthogonal Matching Pursuit ◽  
Sparse Signals

AbstractSign truncated matching pursuit (STrMP) algorithm is presented in this paper. STrMP is a new greedy algorithm for the recovery of sparse signals from the sign measurement, which combines the principle of consistent reconstruction with orthogonal matching pursuit (OMP). The main part of STrMP is as concise as OMP and hence STrMP is simple to implement. In contrast to previous greedy algorithms for one-bit compressed sensing, STrMP only need to solve a convex and unconstrained subproblem at each iteration. Numerical experiments show that STrMP is fast and accurate for one-bit compressed sensing compared with other algorithms.

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