1,721,144 research outputs found
On Bernstein-Schnabl operators on the unit interval
No full textBernstein-Schnabl operators were first introduced by
R. Schnabl in 1968 in the context
of sets of probability Radon measures on compact Hausdorff spaces.
Subsequently Grossman proposed a general method
of constructing Bernstein-Schnabl operators on an arbitrary convex
compact subset of a locally convex space and he showed that they
are an approximation process for continuous functions.
A particular class of these operators has been also studied by the
F. Altomare and,
subsequently, by several other authors. Their construction
essentially involves positive projections and they satisfy many additional
properties useful for the study of evolution problems.
In this paper we deep the study of the Bernstein-Schnabl
operators associated with a general continuous selection of
probability Borel measures on the interval [0,1], which not
necessarily arise from a positive projection. These operators seem
to have some interest because they furnish new general
approximation processes for continuous functions and they also
approximate the solutions of the initial-boundary problems
associated with a class of degenerate diffusion equations.
In the first section we recall their definition and discuss some
examples of them. After that, we investigate their approximation
properties and show several estimates of the rate of convergence
by means of suitable moduli of smoothness.
Shape preserving properties are discussed in Section 2.
In particular, we investigate some conditions under which these
operators preserve the convexity.
In the third section we show that suitable iterates of
Bernstein-Schnabl operators converge to a Markov semigroup on
C([0,1]) whose generator is a degenerate differential
operator of the form Au(x):=\alpha(x) u''(x) (x \in [0,1]
defined on a suitable subspace of smoot functions satisfying the so-called Wentcel boundary conditions.
By means of Bernstein-Schnabl operators we establish some
qualitative properties of this semigroup and, in particular, its
asymptotic behaviour.
In the same section we also study the generation properties of
general differential operators and
determine suitable continuous selections of Borel measures such
that the iterates of the corresponding Bernstein-Schnabl operators
converge to the given Markov semigroup
Korovkin-type Approximation Theory and its Applications
No full textThe power of the original result by Korovkin impressed many mathematicians and hence a considerable amount of research extended this theorem to the setting of different function spaces or more general abstract spaces such as Banach lattices, Banach algebras, Banach spaces and so on. At the same time, strong and fruitful connections of this theory have also been revealed not only with classical approximation theory, but also with other fields such as functional analysis (abstract Choquet boundaries and convexity theory, uniqueness of extensions of positive linear forms, convergence of sequences of positive linear operators in Banach lattices, structure theory of Banach lattices, convergence of sequences of linear operators in Banach algebras and in C*-algebras, structure theory of Banach algebras, approximation problems in function algebras), harmonic analysis (convergence of sequences of convolution operators on function spaces and function algebras on (locally) compact topological groups, structure theory of topological groups), measure theory and probability theory (weak convergence of sequences of positive Radon measures and positive approximation processes constructed by probabilistic methods), and partial differential equations (approximation of solutions of Dirichlet problems and of diffusion equations).
This work, in fact, delineated a new theory called Korovkin-type approximation theory.
The reader will find a quite complete picture of what has been achieved in the field, a modern and comprehensive exposition of the main aspects of Korovkin-type approximation theory in spaces of continuous real functions together with its main applications. The function spaces we have chosen to treat play a central role in the whole theory and are the most useful for the applications in the various univariate, multivariate and infinite dimensional settings.
The book is mainly intended as a reference text for research workers in the field; a large part of it can also serve as a textbook for a graduate level course. The organization of the material does not follow the historical development of the subject and allows us to present the most important part of the theory in a concise way. Chapters 2, 3 and 4 are devoted to the main aspects of Korovkin-type approximation theory in C_0(X) and C(X)-spaces. In Chapter IV we also point out the strong interplay between KAT and Choquet's integral representation theory, as well as Stone-Weierstrass-type theorems.
Chapters 5 and 6 are mainly concerned with applications to: Approximation of continuous functions by means of positive linear operators, Approximation and representation of the solutions of particular partial differential equations of diffusion type, by means of powers of positive linear operators, More precisely, in Chapter 5 we give the first and best-known applications of Korovkin-type approximation theory. We describe different kinds of positive approximation processes. Particular care is devoted to probabilistic-type operators, discrete-type operators, convolution operators for periodic functions and summation methods. In the final Chapter 6 we present a detailed analysis of some further sequences of positive linear operators that have been studied recently. These operators play an important role in some fine aspects of approximation theory. They connect the theory of C_0-semigroups of operators, partial differential equations and Markov processes. The main examples we consider are the Bernstein-Schnabl operators, the Stancu-Schnabl operators and the Lototsky-Schnabl operators.
Subsequently we show how these operators are strongly connected with initial and (Ventcel-type) boundary value problems in the theory of partial differential equations.
Although the aim of the book is to survey both classical and recent results in the field, the reader will find a certain amount of new material. In any case, the majority of the results presented here appears in a book for the first time
Undeserved authorship in surgical research: an underestimated bias with potential side effects on academic careers
No full text
Bernstein-Schnabl operators on noncompact real intervals
No full textWe study the Bernstein-Schnabl operators associated with a continuous selection of probability Borel measure on a noncompact real interval in the framework of weighted spaces of continuous functions.
We investigate their approximation properties and, in addition, we prove that their iterates converge to a positive C_0-semigroup whose generator is a differential operator of the form Au := αu′′. A converse problem on the half line is also discussed
On a generalization of Szasz-Mirakjan-Kantorovich operators
No full textIn this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L p ([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szász–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness
Cores for second-order differentiali operators on real intervals
No full textWe investigate several general conditions in order to determine some cores for gen- erators of strongly continuous positive semigroups of the form Au := αu′′ on weighted spaces of continuous functions on an arbitrary noncompact real interval. As an application we consider a degenerate differential operator of the above mentioned form on the interval [0, +∞[ and we estab- lish an approximation formula for the corresponding positive semigroup in terms of iterates of an integral modification of Sza ́sz-Mirakjan operators
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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