Positive Linear Operators And Semigroups Of Operators

One of main topics of Approximation theory is the study of positive linear op-erators.Since the classical Bernstein operators were proposed by S.N.Bernstein in 1912,approximating continuous functions by polynomial operators has more than one hundred years history.Classical Bernstein operators not only find their applications in Approximation theory but also in computing disciplines.In recent decades,clas-sical Bernstein operators and Bernstein basis functions also play an important role in Computer Aided Geometric Design(CAGD).Especially the Bernstein basis functions find wide applications in geometric modeling,and again stimulated research interest of these operators.In general,we use classical Bernstein operators to approximate continuous functions on finite closed interval,and use Szász-Mirikan operators to ap-proximate unbounded continuous functions on infinite interval.The main drawback of Szász-Mirikan operators is that these operators are infinite series,in applications it is quite hard to deal with infinite series;instead,we usually use a partial sum of Szász-Mirikan operators.However,when the parameters are conveniently chosen,classical Bernstein operators can also approximate unbounded functions on infinite intervals.In this thesis,we shall first construct semigroup structural representations for many kinds of Bernstein-type operators,and by using these representations and semigroup theory uncover some new essential properties of these Bernstein-type operators.Classical Bernstein operators have many important generalizations.Two rep-resentative examples are q-Bernstein operators and Chebyshev-Bernstein operators.Another generalization are Lototsky-Bernstein operators.These operators were first proposed by King in 1965,and then in the 1970's,Eisenberg and Wood considered uniform approximation of analytic functions by Lototsky-Bernstein operators.Since then,research on these operators has been completely ignored.Since the 1980's,poly-nomial blossoming has been widely used in CAGD;blossoming also has many advan-tages in research on B-spline.Classical Bernstein operators are polynomial function of order n.By applying blossoming to the Bernstein operators and then replacing ev-ery new variable of n multi-variables by an increasing functions pi(x),i ? 1,we get the Lototsky-Bernstein operators.In this thesis,we shall systematically study these operators.In Chapter 2,we shall discuss the convergence of approximating Szász operators by using many kinds of Bernstein-type operators.The main analysis tools is the appli-cations of semigroups of operators.By constructing semigroup structural representa-tions of Bernstein-type operators and using semigroup theory,we can improve many classical results on this topic.We also consider the convergence of Szász-Durrmeyer operators by Bernstein-Durrmeyer operators.In the final part of this Chapter,we in-troduce the Shorgin formula to study Bernstein operators.We find that by using this Shrogin formula,we can uncover some important essential properties of Bernstein operators which have never before been discussed.For example we can use classi-cal Bernstein operators to approximate unbounded functions on infinite interval.In Chapter 3,we shall also use the Shorgin formula to deduce asymptotic properties of Lototsky-Bernstein operators.In Chapter 3,we shall systematically study the shape preserving properties of Lototsky-Bernstein operators,including fixed points theory?approximation prop-erties of fixed points?iterations?bounded variation diminishing properties?total positivity of Lototsky-Bernstein basis functions and variation diminishing proper-ties?shape preserving propeties(monotonicity preserving,convexity preserving),de-pendence of pi(x)and asymptotic convergence.We must especially emphasize the total positivity of Lototsky-Bernstein basis functions.Normalized and totally positive basis functions are quite suitable for geometric design.We shall touch on this topic in our future work.In Chapter 4,we shall mainly discuss the Lototsky-Bernstein operators when all the fixed points ?np(x)are identical.In general,the fixed point functions ?np(x)depend on pi(x),1 ? i ? n.;even when all the pi(x)are identical,the fixed points functions ?np(x)may also be different from each other.Two problems are how can we restrict these pi(x)to guarantee all the ?np(x)are identical,and what properties will appear when all the ?np(x)are identical.In this Chapter,we shall totally resolve these issues.At the end of this Chapter,we determine conditions on p1(x)ensuring all the Pj(x)(j>2)are increasing as well and the Lototsky-Bernstein operators Ln(f;x)are monotonic in n just as in the case of the classical Bernstein operators when f is(1,p1)-convex.