Positive operators and approximation in function spaces on completely regular spaces.
Preconditioners and Korovkin-type theorems for infinite-dimensional bounded linear operators via completely positive maps
The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and...
Projection solutions of Frobenius-Perron operator equations.
Properties of the modulus of continuity for monotonous convex functions and applications.
-parametric Bleimann Butzer and Hahn operators.
Quantitative Krovkin Type Theorems on Simultaneous Approximation.
Rate of approximation for Bézier variant of Bleimann-Butzer and Hahn operators.
Rate of approximation for the Bézier variant of Balazs-Kantorovich operators
Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation.
Rate of convergence of Chlodowsky operators for functions with derivatives of bounded variation.
Regular vector lattices of continuous functions and Korovkin-type theorems-Part II
By applying the results of the first part of the paper, we establish some Korovkin-type theorems for continuous positive linear operators in the setting of regular vector lattices of continuous functions. Moreover, we present simple methods to construct Korovkin subspaces for finitely defined operators and for the identity operator and we determine those classes of operators which admit finite-dimensional Korovkin subspaces. Finally, we give a Korovkin-type theorem for continuous positive projections....
Remarks on an article of J.P. King
The present note discusses an interesting positive linear operator which was recently introduced by J.P. King. New estimates in terms of the first and second modulus of continuity are given, and iterates of the operators are considered as well. For general King operators the second moments are minimized.
Remarks on Voronovskaya's theorem.
Representation formulae for (C₀) m-parameter operator semigroups
Some general representation formulae for (C₀) m-parameter operator semigroups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C₀) one- and m-parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.
Representation formulas for cosine and sine functions of operators II.
Representation formulas for C-semigroups.
Riesz's lattices and almost positive operators.
Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator
The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the...
Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions
We introduce and study a one-parameter class of positive linear operators constituting a link between the well-known operators of S. N. Bernstein and their genuine Bernstein-Durrmeyer variants. Several limiting cases are considered including one relating our operators to mappings investigated earlier by Mache and Zhou. A recursion formula for the moments is proved and estimates for simultaneous approximation of derivatives are given.
Simultaneous approximation by linear combination of modified bernstein polynomials