Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

The aim of this paper is the study of a new sequence of positive linear approximation operators M_n, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.     

Approximation gewisser Operatoren durch Operatoren endlichen Ranges und eine Kollokationsmethode

Summary Certain bounded linear operators from Hilbert spaces H into function spaces C(X) are approximated by operators of finite rank with the aid of orthogonal projections. As examples for such operators we present special integral operators and linear partial differential operators. As application we consider a collocation method.     

Approximation by Durrmeyer–Bezier operators

In the present paper, we consider the Bezier variant M_n,α(f,x) of the generalized Durrmeyer type operators, and obtain an estimate on the rate of convergence of M_n,α(f,x) for the decomposition technique of functions of bounded variation. In the end we propose an open problem for the readers and give an asymptotic formula for these generalized Durrmeyer type operators.     

The Integral Type Modification of Jain Operators and its Approximation Properties

Abstract

In the present paper, we discuss the approximation properties of Durrmeyer-Stancu type variant of Jain operators with the modified forms of the Beta basis functions. We establish some direct results, which include the asymptotic formula, the error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct a King modification of these operators which preserves the test functions e₀ and e₁.     

A Factorization Property of R‐Bounded Sets of Operators on Lp‐Spaces

Let T be a bounded operator on L^p‐space, with 1 ≤ p < ∞. A theorem of W. B. Johnson and L. Jones asserts that after an appropriate change of density, T actually extends to a bounded operator on L². We show that if 𝒯 ⊂ B (L^p) is an R‐bounded set of operators, then the latter result holds for any T ∈ 𝒯 with a common change of density. Then we give applications including results on R‐sectorial operators.     

Applications of B‐bounded nonlinear semigroups to particle transport problems

In this paper we study an application of nonlinear B‐bounded semigroups introduced in a previous paper. The application is similar to the particle transport problem which led to B‐bounded linear semigroups. We deal with a nonlinear particle transport problem, which can be solved by using B‐bounded nonlinear semigroups. Copyright © 2010 John Wiley & Sons, Ltd.     

Polynomial Trace Estimates for Semigroups

For semigroups (e ^tA ) _t0 of operators on a Hilbert space, we give conditions guaranteeing trace estimates of the polynomial type 0$$ " align="middle" border="0"> , where denotes the trace class. As an application we present higher order analogues of results due to E.B. Davies, B. Simon and M. van den Berg of the type 0$$ " align="middle" border="0"> , for certain unbounded domains , e.g. spiny urchin domains.     

U-ideals of factorable operators

We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons's construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are u-ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.     

Approximation properties for King's type modified q‐Bernstein–Kantorovich operators

In the present research article, we introduce the King's type modification of q‐Bernstein–Kantorovich operators and investigate some approximation properties. We show comparisons and present some illustrative graphics for the convergence of these operators to some function. Copyright © 2015 John Wiley & Sons, Ltd.     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Spline approximation of a non-linear Riemann–Hilbert problem†

In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H ¹(Γ) → H ¹(Γ) is defined via a Nemytski operator. We approximate A by A ⁿ : H ¹(Γ) → H ¹(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. ?nirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A ⁿ (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation.     

On Measuring the Efficiency of Kernel Operators in L p (R d )

It is well known that it is possible to enhance the approximation properties of a kernel operator by increasing its support size. There is an obvious tradeoff between higher approximation order of a kernel and the complexity of algorithms that employ it. A question is then asked: how do we compare the efficiency of kernels with comparable support size? We follow Blu and Unser and choose as a measure of the efficiency of the kernels the first leading constant in a certain error expansion. We use time domain methods to treat the case of globally supported kernels in L _p (R ^d), 1≤p≤∞.

     

Approximation Properties of the Operators ⋎n+2r(f) and of Their Discrete Analogs

This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier--Legendre sums of order n with 2r terms of the form _k=1 ^2r a_kP_n+k(x) added; here P _m(x) denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval [-1,1], which, in fact, for r= = 1 allows us to significantly improve the approximation properties of partial Fourier--Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions and A _q (B). With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties.     

Mosaic approximations of discrete analogs of Calderón-Zygmund operators

Asymptotic estimates of the form mrA = O(InN · ln ^d ɛ ⁻¹), whered is the dimension of the initial space, for mosaic ranks of discrete analog of Calderón-Zygmund operators are obtained for various mosaic covers. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 81–94, January, 1998.     

Construction of a new family of Bernstein‐Kantorovich operators

In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first‐ and second‐order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein‐Kantorovich operators and their approximation behaviors.     

On the convergence of the product of independent random operators

In this paper, we study infinite products of independent random operators. Conditions for the existence a.s. and the convergence in mean of order p of such infinite products are established.     

On a generalization of the extended best polynomial approximation operator in Orlicz–Lorentz spaces

In this paper, we consider the best polynomial approximation operator defined on an Orlicz–Lorentz space ${\mathrm{\Lambda }}_{w,\varphi }$, and its extension to ${\mathrm{\Lambda }}_{w,{\varphi }^{\prime }}$, where w is a non-negative continuous weight function and ${\varphi }^{\prime }$ is the derivative of ϕ, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz–Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation.     

An answer to Hermann's conjecture on Bleimann–Butzer–Hahn operators

We give a negative answer to the conjecture of Hermann [On the operator of Bleimann, Butzer and Hahn, in: J. Szabados, K. Tandori (Eds.), Approximation Theory, Proc. Conf., Kecskemét/Hung., 1990, North-Holland Publishing Company, Amsterdam, 1991, Colloq. Math. Soc. János Bolyai 58 (1991) 355–360] on Bleimann–Butzer–Hahn operators L_n. Our main result states that for each locally bounded positive function h there exists a continuous positive function f defined on [0,∞) with L_nf→f(n→∞), pointwise on [0,∞), such that

Moreover we construct an explicit counterexample function to Hermann's conjecture.     

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r^th order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.