Composition and Decomposition of Positive Linear Operators (VIII) (original) (raw)
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On the composition and decomposition of positive linear operators (VII)
Applicable Analysis and Discrete Mathematics, 2021
In the present paper we study the compositions of the piecewise linear interpolation operator S∆ n and the Beta-type operator Bn, namely An := S∆ n •Bn and Gn := Bn • S∆ n. Voronovskaya type theorems for the operators An and Gn are proved, substantially improving some corresponding known results. The rate of convergence for the iterates of the operators Gn and An is considered. Some estimates of the differences between An, Gn, Bn and S∆ n , respectively, are given. Also, we study the behaviour of the operators An on the subspace of C[0, 1] consisting of all polygonal functions with nodes 0, 1 2 ,. .. , n−1 n , 1. Finally, we propose to the readers a conjecture concerning the eigenvalues of the operators An and Gn. If true, this conjecture would emphasize a new and strong relationship between Gn and the classical Bernstein operator Bn.
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Following a 1939 article of Favard we consider the composition of classical Bernstein operators and piecewise linear interpolation at mutually distinct knots in [0, 1], not necessarily equidistant. We prove direct theorems in terms of the classical and the Ditzian-Totik modulus of second order. 2010 Mathematics Subject Classification: 41A10, 41A15, 41A17, 41A25, 41A36. Key words and phrases: Favard-Bernstein operator, Bernstein polynomials for arbitrary points of interpolation, piecewise linear interpolation, Bernstein operator, second order modulus of smoothness, Ditzian-Totik modulus, positive linear operator, degree of approximation.
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2012
The central problem in this technical report is the question if the classical Bernstein operator can be decomposed into nontrivial building blocks where one of the factors is the genuine Beta operator introduced by M\"uhlbach and Lupa\c{s}. We collect several properties of the Beta operator such as injectivity, the eigenstructure and the images of the monomials under its inverse. Moreover, we give a decomposition of the form Bn=barmathbbBncircFnB_n = \bar{\mathbb{B}}_n \circ F_n Bn=barmathbbBncircFn where FnF_nFn is a nonpositive linear operator having quite interesting properties. We study the images of the monomials under FnF_nFn, its moments and various representations. Also an asymptotic formula of Voronovskaya type for polynomials is given and a connection with a conjecture of Cooper and Waldron is established. In an appendix numerous examples illustrate the approximation behaviour of FnF_nFn in comparison to BnB_nBn.
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2014
Of concern is the study of the eigenstructure of some classes of positive linear operators satisfying particular conditions. As a consequence, some results concerning the asymptotic behaviour as \(t\to +\infty\) of particular strongly continuous semigroups \((T(t))_{t\geq 0}\) expressed in terms of iterates of the operators under consideration are obtained as well. All the analysis carried out herein turns out to be quite general and includes some applications to concrete cases of interest, related to the classical Beta, Stancu and Bernstein operators.
Products of Positive Operators
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On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class {\mathcal {L}^{+\,2}}L+2ofboundedoperatorsonseparableinfinitedimensionalHilbertspaceswhichcanbewrittenastheproductoftwoboundedpositiveoperatorsisstudied.Thestructureismuchricher,andconnects(butisnotequivalentto)quasi−similarityandquasi−affinitytoapositiveoperator.ThespectralpropertiesofoperatorsinL + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators inL+2ofboundedoperatorsonseparableinfinitedimensionalHilbertspaceswhichcanbewrittenastheproductoftwoboundedpositiveoperatorsisstudied.Thestructureismuchricher,andconnects(butisnotequivalentto)quasi−similarityandquasi−affinitytoapositiveoperator.Thespectralpropertiesofoperatorsin{\mathcal {L}^{+\,2}}L+2aredeveloped,andmembershipinL + 2 are developed, and membership inL+2aredeveloped,andmembershipin{\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.
Existence and extensions of positive linear operators
Positivity, 2008
In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied.
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Integral Equations and Operator Theory, 1996
We characterize when positive operators can be factored by "analytic Toeplitz" type operators. As a corollary, we give an operator theory characterization of those invariant subspaees of doubly commuting unilateral shifts, which are generated by a single inner function on the bidisk. The last result extends to shifts of arbitrary (countable) multiplicity.