Voronovskaja type theorems for positive linear operators related to squared Bernstein polynomials

We provide an analysis of the rate of convergence of positive linear approximation operators defined on C[0, 1]. We obtain a sufficient condition for a sequence of positive linear approximation operators to possess a Mamedov-type property and give an application to the Durrmeyer approximation process.

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

It is proved that at almost all points the order of approximation, even of one of the functions 1, cos x,sin x by means of a sequence of linear positive polynomial operators having uniformly bounded norms, is not higher than 1/n2. Refinements of this result are given for operators of convolution type.

Let [a,b] be a compact interval of the real line IR. It is known that the classical theorem of Bohman-Korovkin states that in order that a sequence of positive linear operators (Lm), mapping into itself the space C[a,b] of continuous real-valued functions on [a,b], equipped with the uniform norm, to have the property that, for any f∈C[a,b], if m → ∞ we have lim Lmf = f, uniformly on [a,b], it is necessary and sufficient that such a convergence occur for a triplet of “test functions” from C[a,b], forming a so called Korovkin system. For C[a,b] the three monomials e0,e1,e2 where ej(x) := xj (j = 0,1,2), represent such a system.

Here we consider and study very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions. These are acting on the space of real fractionally differentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related fractional stochastic Shisha-Mond type inequalities of \(L^{q}\)-type \(1\le q\,<\infty \) and corresponding fractional stochastic Korovkin type theorems. These are regarding the stochastic q-mean fractional convergence of a sequence of stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are produced with rates and are given via the fractional stochastic inequalities involving the stochastic modulus of continuity of the \(\alpha \)th fractional derivatives of the engaged stochastic process, \(\alpha >0\), \(\alpha \notin \mathbb {N}\). The impressive fact is that the basic real Korovkin test functions assumptions are enough for the conclusions of our fractional stochastic Korovkin theory. We give applications to stochastic Bernstein operators. See also [10].

Let T be a positive linear operator on L1 of a probability space such that II T1j i: 1. In this note we consider the following question: Under what condition is T multiplicative on Let (X, F, m) be a probability space and L1=L1(X, F, m) the Banach space of equivalence classes of integrable complex-valued functions on X. Let T be a positive linear operator on L1 such that II TIIl< 1, and define G(T)={f E L1; If I = I Tf I = 1). The main purpose of this note is to prove the following: THEOREM. If G(T) is total in L1, i.e., the linear manifold generated by G(T) is dense in L1 in the norm topology, then for any bounded functions f and g we have T(fg)= (Tf)(Tg). As a corollary of the Theorem we have the following result; essentially the same idea has been used by Halmos [2, p. 45] to find a necessary and sufficient condition for a unitary operator on L2=L2(X, Y, m) to be induced by an automorphism of the measure algebra (F(m), m) associated with (X, F, m). COROLLARY. A necessary and sufficient condition that T be induced by a homomorphism of the measure algebra (f(m), m) into itself is that G(T) be total in L1. For the proof of the theorem we need some lemmas. LEMMA 1. Let K be a compact Hausdorff space and C(K) the Banach algebra of all complex-valued continuous functions f on K wvith norm lIf IIU=sup{If(s)I; s e K}. Let U be a positive linear operator on C(K) such that U1=1. Let f,geC(K), IfI=lgi=1, and IUfI=IUgi=1. Then U(fg)=(Uf)(Ug)Received by the editors February 28, 1973 and, in revised form, May 10, 1973. AMS (MOS) subject classifications (1970). Primary 47B55, 47A35; Secondary 28A65, 46J10.

ABSTRACTThe Mihesan operator is the generalization of well‐known positive linear operators like Bernstein operators, Szász–Mirakjan operators, Baskakov operators, and Lupaş operators. In order to expand the theory of positive operators beyond the linear example, nonlinear positive operators are essential. We have introduced the nonlinear Mihesan operators and find the approximation properties of Truncated Mihesan operators with all steps in this note.

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space Cρ1 into a weighted space Bρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.

In this paper, following very recent results of Baliarsingh (Alex Eng J 55(2):1811–1816, 2016), we first introduce the concepts of statistically $$\Omega ^{\Delta }$$ -summability and $$\Omega ^{\Delta }$$ -statistical convergence by means of fractional-order difference operator $$\Delta ^{\alpha ,\beta ,\gamma }_{h}$$ . We also present some important inclusion relations between newly proposed methods. Our present investigation deals essentially with various summability techniques and reveals how these methods lead to a number of approximation by positive linear operators. As an application, we prove a Korovkin type approximation theorem and also present an illustrative example using the generating function type Meyer-Konig and Zeller operator. Furthermore, we estimate the rate of convergence of approximating linear operators by means of the modulus of continuity and some Voronovskaja type results are derived. Finally, we present some computational and geometrical interpretations to illustrate some of our approximation results.

In this paper, we use the power series methods to study Korovkin-type approximation theorems for sequences of operators that are monotone and sublinear. It is important to point out that any positive linear operator is monotone sublinear but the converse is not true. We also calculate the rate of convergence in terms of modulus of continuity. Finally, we provide some examples that satisfy our theorems and give quantitative estimates. We also define Bernstein-Chlodovsky-Kantorovich-Choquet (BCKC) polynomial operators similar to those given before and obtain approximation estimate.

Recently there has been a great amount of research both qualitative and quantitative on the problem of approximating functions with certain growth conditions by means of linear positive operators. This paper discusses a quantitative problem for unbounded functions, i.e., the order of approximation to unbounded functions by means of linear positive operators. The theorems presented are, in several instances, extensions of known results from the case of bounded functions to the unbounded case. Applications of these theorems are given to spline functions, which are presently the subject of intensive investigation, and to the well-known Szasz-Mirakyan-Hille operators.

Iteration of positive approximation operators

We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.

We define a general linear operator from which we obtain as special case the modified beta first kind operator\[(B_{p,q}f)(x)={B(p,q)}\int_0^1 t^{p-1}(1-t)^{q-1}f\left({B(p,q)}{B(p+a,q)}t^a x\right){\rm d}t.\]We consider here only the cases \(a=1\) and \(a=-1\).We obtain several positive linear operators as particular cases of this modified beta first kind operator.

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an rth order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.