Approximation by a new positive linear operator with exponential weight function

As far as we know, the main drawback of the classical positive linear operators of a given function f (under suitable conditions) is that they are of order one. Thus they converge to f very slowly. This paper presents a new class of integral-type positive linear operators that are of order one. Additionally, a second-order improvement of the operator is also provided. Uniform convergence, asymptotic order of convergence and other important properties are proved in detail. Two discrete forms of the operators suitable for numerical purposes are also introduced. Several examples are given in the end to verify the theoretical results numerically.

In this note we prove that an estimate due to Ivanov and Pichugov, for positive linear convolution operators in Lp spaces, can be extended to a class of positive linear operators which are not of convolution type.

We study inequalities involving convex functions and positive linear operators and interpret them within the framework of convex stochastic order ≤ cx . Our approach is operatorial in nature and provides a unified view of such inequalities involving several classical approximation processes. By considering a broad class of positive linear operators, we obtain new inequalities that describe dominance between the corresponding induced measures. In addition to the direct problem, showing that the convexity of a function f implies inequalities of the type L n f ≥ M n f , we also consider the inverse problem: given f and a suitable family of measures μ ≤ cx ν satisfying ∫ f d μ ≤ ∫ f d ν , does it follow that f must be convex? This inverse problem is particularly challenging. As an illustrative example, we ask whether, for f ∈ C [0, ∞), the condition S t f ≤ V t f for all t > 0, where S t and V t denote the Szász–Mirakjan and classical Baskakov operators respectively, necessarily implies the convexity of f .

We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein-Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the derivatives of the operator images are included. Our main tool is the eigenstructure of the members of the class.

Unbounded functions and positive linear operators

A note on the fixed points of positive linear operators

A unified class of linear positive operators has been defined. Using these operators some approximation estimates have been obtained for unbounded functions. For particular linear positive operators these results sharpen and improve the earlier estimates due to Fuhua Cheng (J. Approx. Theory, 1984) and Xiehua Sun (J. Approx. Theory, 1988).

We introduce a class of positive linear operators defined by Steklov means, investigate their properties and prove that the Weierstrass operators can be approximated in terms of the Steklov averages.

Approximation theorems for certain positive linear operators

We study power series of members of a class of positive linear operators reproducing linear function constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach.

Let (X, 6B, m) be a finite measure space. We shall denote by Lv(m) (1 ?p < oo) the Banach space of all real-valued 63-measurable functionsf defined on X such that if I P is m-integrable, and by Lo?(m) the Banach space of all real-valued, 63-measurable, m-essentially bounded functions defined on X; as usual, the norm in LP(m) is given by lIpIfI = {fxlf(x)IPdm}1/P, and the norm in Lcr(m) by IIg||OO -m-ess. supzEEx lg(x)l. Two functions in LP(m) or Lo((m) will be identified if they differ only on a set of m-measure zero. In this note, we shall be concerned with a positive linear operator T of L'(m) into L1(m) with III1Ti<1. We say that the pointwise ergodic theorem (the L'(m)-mean ergodic theorem, respectively) holds for such an operator T if for every f in L'(m), the sequence of averages { 1 (/n) k=0 Thf } converges m-almost everywhere (in the norm of L'(m), respectively) to a function in Ll(m). Recently, R. V. Chacon [1] constructed a class of positive linear operators in L1(m) with the norm equal to 1 for which the pointwise ergodic theorem fails to hold. Also, A. Ionescu Tulcea [5], [61 showed that in the group of all positive invertible linear isometries of L(r(m) the set of all T for which the pointwise ergodic theorem fails to hold forms a set of second category with respect to the strong operator topology. On the other hand, the ergodic theorem of HopfDunford-Schwartz [4] tells us that if, in addition, T maps L(r(m) into Lco (m) and IIT| < 1, then the pointwise ergodic theorem is valid for such T. In view of these facts, it is interesting to find out what other additional conditions on T would guarantee the validity of the pointwise ergodic theorem. In this note, we shall find a few such conditions which are weaker than the condition of the Hopf-Dunford-Schwartz theorem (though our conditions seem to work for a finite measure space only). We also obtain a result (corollary to Theorem 1, below) which generalizes a result obtained by N. Dunford and D. S. Miller in [3]. First of all, let us observe that if our operator T satisfies ITI1 <1, then the pointwise ergodic theorem is always valid. This is because, for such an operator T, nI Tnf(x) I < rnm-almost everywhere for every f in L' (m), since

In this paper we demonstrate a general property for a class of linear positive operators. By particularization, we obtain the convergence and the evaluation for the rate of convergence in term of the first modulus of smoothness for the Bernstein operators, Durrmeyer operators, Kantorovich operators and Bleimann, Butzer and Hahn operators.

An asymptotic relation is established between the rate of approximation of functions from space $$C^2[0,1]$$ by a general class of positive linear operators and a generalized second order Ditzian–Totik modulus. In addition, it is shown by a counterexample that this asymptotic relation is not true for general continuous functions.

On statistical approximation of a general class of positive linear operators extended in q-calculus

On the approximation of functions in [formula omitted] by a class of positive linear operators