Riesz-type representation for positive linear operators preserving continuity

A Riesz-type representation theorem is given for all positive linear endomorphisms of the space of continuous functions on a compact interval.     

Sobolev and strict continuity of general hysteresis operators

The most natural and important topologies connected with hysteresis operators are those induced by uniform convergence, W^{1, 1}‐convergence, and strict convergence. Indeed the supremum norm and the variation are invariant under reparameterization. We prove a general result that implies that if a hysteresis operator is continuous with respect to the topology of W^{1, 1}, then it is continuous with respect to the strict topology. Copyright © 2009 John Wiley & Sons, Ltd.     

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r^th order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.     

Greedy wavelet projections are bounded on BV

Let be the space of functions of bounded variation on with . Let , , be a wavelet system of compactly supported functions normalized in , i.e., , . Each has a unique wavelet expansion with convergence in . If is the set of indicies for which are largest (with ties handled in an arbitrary way), then is called a greedy approximation to . It is shown that with a constant independent of . This answers in the affirmative a conjecture of Meyer (2001).

     

Hankel-type operators on the space of bounded harmonic functions

We study Hankel-type operators on the space of bounded harmonic functions on the open unit disk. These operators are related to tight uniform algebras, the Dunford-Pettis property, and Bourgain algebras.

     

Approximation by Durrmeyer–Bezier operators

In the present paper, we consider the Bezier variant M_n,α(f,x) of the generalized Durrmeyer type operators, and obtain an estimate on the rate of convergence of M_n,α(f,x) for the decomposition technique of functions of bounded variation. In the end we propose an open problem for the readers and give an asymptotic formula for these generalized Durrmeyer type operators.     

On the integrated Baskakov type operators

The present paper deals with the study of the rate of convergence of the integrated Baskakov-Durrmeyer type operators in simultaneous approximation. We also mention some of the improvements on the recent paper of Deo [N. Deo, Direct results on exponential type operators, Applied Mathematics and Computation 204(1) (2008) 109-115].     

A certain family of summation-integral type operators

In the present paper, the authors introduce and investigate a new sequence of linear positive operators G_n,c, which includes some well-known operators as its special cases. The results obtained here include an estimate on the rate of convergence of G_n,c (f, x) by means of the decomposition technique for functions of bounded variation.     

The play operator on the rectifiable curves in a Hilbert space

The vector play operator is the solution operator of a class of evolution variational inequalities arising in continuum mechanics. For regular data, the existence of solutions is easily obtained from general results on maximal monotone operators. If the datum is a continuous function of bounded variation, then the existence of a weak solution is usually proved by means of a time discretization procedure. In this paper we give a short proof of the existence of the play operator on rectifiable curves making use of basic facts of measure theory. We also drop the separability assumptions usually made by other authors. Copyright © 2007 John Wiley & Sons, Ltd.     

Hypercyclic composition operators on Hv0‐spaces

We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator C_φ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators C_φ on $H_{v}^0$.     

Degree of approximation to function of bounded variation by Bézier variant of MKZ operators

Guo (Approx. Theory Appl. 4 (1988) 9-18) introduced the integral modification of Meyer-Konig and Zeller operators and studied the rate of convergence for functions of bounded variation. In this paper we introduce the Bézier variant of these integrated MKZ operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of MKZ basis functions. Recently, Zeng (J. Math. Anal. Appl. 219 (1998) 364-376) claimed to improve the results of Guo and Gupta (Approx. Theory Appl. 11 (1995) 106-107), but there is a major mistake in the paper of Zeng. For special case our main theorem gives the correct estimate on the rate of convergence, over the result of Zeng.     

Approximation by bicomplex beta operators in compact ‐disks

The main object of this paper is to extend some known approximation results for the complex beta operators of the first kind attached to analytic functions in strips of compact disks. The extensions of these approximation properties, which are presented here, involve bicomplex beta operators in compact ‐disk. Several other related results are also considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

A Factorization Property of R‐Bounded Sets of Operators on Lp‐Spaces

Let T be a bounded operator on L^p‐space, with 1 ≤ p < ∞. A theorem of W. B. Johnson and L. Jones asserts that after an appropriate change of density, T actually extends to a bounded operator on L². We show that if 𝒯 ⊂ B (L^p) is an R‐bounded set of operators, then the latter result holds for any T ∈ 𝒯 with a common change of density. Then we give applications including results on R‐sectorial operators.     

Exponential rate of convergence for some Markov operators

The exponential rate of convergence for Markov operators is established. The operators correspond to continuous iterated function systems which are a very useful tool in some cell cycle models.     

On convergence of general Gamma type operators

The present paper deals with the new type of Gamma operators, here we esti- mate the rate of pointwise convergence of these new Gamma type operators Mn,k for func- tions of bounded variation, by using some techniques of probability theory.     

Hahn-Banach extension operators and spaces of operators

Let be Banach spaces and let be closed operator ideals. Let be a Banach space having the Radon-Nikodým property. The main results are as follows. If is a Hahn-Banach extension operator, then there exists a set of Hahn-Banach extension operators , , such that , where . If is an ideal in for all equivalently renormed versions of , then there exist Hahn-Banach extension operators and such that .

     

Discrete operators of sampling type and approximation in ‐variation

The paper deals with approximation results with respect to the φ‐variation by means of a family of discrete operators for φ‐absolutely continuous functions. In particular, for the considered family of operators and for the error of approximation, we first obtain some estimates which are important in order to prove the main result of convergence in φ‐variation. The problem of the rate of approximation is also studied. The discrete operators that we consider are deeply connected to some problems of linear prediction from samples in the past, and therefore have important applications in several fields, such as, for example, in speech processing. Moreover such family of operators coincides, in a particular case, with the generalized sampling‐type series on a subset of the space of the φ‐absolutely continuous functions: therefore we are able to obtain a result of convergence in variation also for the generalized sampling‐type series. Some examples are also discussed.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.     

A variational characterization of rate‐independent evolution

Rate‐independent evolution driven by non‐convex potentials is by nature non‐smooth and some weak solvability notions have been recently advanced. This note is intended to contribute to this discussion by proposing a variational characterization of rate‐independent evolution based on a variational principle and a maximal dissipation criterion. The resulting novel solution notion is assessed in an elementary yet critical scalar case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)