Local Approximation Properties Research Articles

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

The objective of this paper is to construct univariate and bivariate blending type α-Schurer–Kantorovich operators depending on two parameters α∈0,1 and ρ>0 to approximate a class of measurable functions on 0,1+q,q>0. We present some auxiliary results and obtain the rate of convergence of these operators. Next, we study the global and local approximation properties in terms of first- and second-order modulus of smoothness, weight functions, and by Peetre’s K-functional in different function spaces. Moreover, we present some study on numerical and graphical analysis for our operators.

A New Generalization of Szász-Mirakjan Kantorovich Operators for Better Error Estimation

In this article, we construct a new sequence of Szász-Mirakjan-Kantorovich operators denoted as $K_{n,\gamma}(f;x)$, which depending on a parameter $\gamma$. We prove direct and local approximation properties of $K_{n,\gamma}(f;x)$. We obtain that, if $\gamma>1$, then the operators $K_{n,\gamma}(f;x)$ provide better approximation results than classical case for all $x\in[0,\infty)$. Furthermore, we investigate the approximation results of $K_{n,\gamma}(f;x)$, graphically and numerically. Moreover, we introduce new operators from $K_{n,\gamma}(f;x)$ that preserve affine functions and bivariate case of $K_{n,\gamma}(f;x)$. Then, we study their approximation properties and also illustrate the convergence of these operators comparing with their classical cases.

Invertible residual networks in the context of regularization theory for linear inverse problems

Abstract Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step procedure. First, one trains the often neural network-based reconstruction scheme via a dataset. Second, one applies the scheme to new measurements to obtain reconstructions. We follow these steps but parameterize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility enables investigating the influence of the training and architecture choices on the resulting reconstruction scheme. For example, assuming local approximation properties of the network, we show that these schemes become convergent regularizations. In addition, the investigations reveal a formal link to the linear regularization theory of linear inverse problems and provide a nonlinear spectral regularization for particular architecture classes. On the numerical side, we investigate the local approximation property of selected trained architectures and present a series of experiments on the MNIST dataset that underpin and extend our theoretical findings.

Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

In this article, we introduce generalized beta extension of Szacute{a}sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

The rate of convergence of a generalization of Post–Widder operators and Rathore operators

In this paper, we study local approximation properties of certain gamma-type operators. They generalize the Post–Widder operators and the Rathore operators, and approximate locally integrable functions satisfying a certain growth condition on the infinite interval [0,infty ). We derive the complete asymptotic expansion for these operators and prove a localization result. Also, we estimate the rate of convergence for functions of bounded variation.

Quasi-interpolation in Spline Spaces: Local Stability and Approximation Properties

In this work we analyze the approximation error in Sobolev norms for quasi-interpolation operators in spline spaces. We establish in a general way the hypotheses on a quasi-interpolant to achieve the optimal order of approximation. Finally, we propose simple but general constructions of such operators that satisfy the established hypotheses and illustrate their performance through some numerical tests.

Asymptotic properties of Kantorovich-type Szász–Mirakjan operators of higher order

In this paper we study local approximation properties of a higher order Kantorovich-type Szász–Mirakjan operator recently introduced by Sabancigil, Kara, and Mahmudov. We derive the complete asymptotic expansion for these operators. They generalize the Szász–Mirakjan operators of Kantorovich-type and approximate locally integrable functions satisfying a certain growth condition on the infinite interval $ {(}0,\infty ) $.

Kantorovich Type Generalization of Bernstein Type Rational Functions Based on (p,q)-Integers

In this paper, we define a new Kantorovich-type (p,q)-generalization of the Balázs–Szabados operators. We derive a recurrence formula, and with the help of this formula, we give explicit formulas for the first and second-order moments, which follow a symmetric pattern. We estimate the second and fourth-order central moments. We examine the local approximation properties in terms of modulus of continuity, we give a Voronovskaja type theorem, and we give the weighted approximation properties of the operators.

Approximation of GBS Type q-Jakimovski-Leviatan-Beta Integral Operators in Bögel Space

In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new operators and then study the degree of approximation by means of modulus of smoothness, mixed modulus of smoothness and Lipschitz class of Bögel continuous functions.

Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

<p style='text-indent:20px;'>The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a} $\end{document}</tex-math></inline-formula>sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.</p>

A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves {1,τ,τ2} instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results.

Some approximation properties of new ( p,q ) -analogue of Bal\xe1zs\u2013Szabados operators

In this paper, a new ( p,q ) -analogue of the Balázs–Szabados operators is defined. Moments up to the fourth order are calculated, and second order and fourth order central moments are estimated. Local approximation properties of the operators are examined and a Voronovskaja type theorem is given.

Finite presentation, the local lifting property, and local approximation properties of operator modules

We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the development of a local theory of operator modules by introducing analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules we prove that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan [37]. We pursue applications to abstract harmonic analysis, where, for a locally compact quantum group G, we show that L1(G)-nuclearity of LUC(G) and L1(G)-semi-discreteness of L∞(G) are both equivalent to co-amenability of G. We establish the equivalence between A(G)-injectivity of G⋉¯M, A(G)-semi-discreteness of G⋉¯M, and amenability of W⁎-dynamical systems (M,G,α) with M injective. We end with remarks on future directions.

Summation-Integral Type Operators Based on Lupas-Jain Functions

We introduce a genuine summation-integral type operators based on Lupaş-Jain type base functions related to the unbounded sequences. We investigated their degree of approximation in terms of modulus of continuity and ????-functional for the functions from bounded and continuous functions space. Furthermore, we give some theorems for the local approximation properties of functions belonging to Lipschitz class. Also, we give Voronovskaja theorem for these operators.

Approximation Properties of p , q -Szász-Mirakjan-Durrmeyer Operators

In this article, we introduce a new Durrmeyer-type generalization of p , q -Szász-Mirakjan operators using the p , q -gamma function of the second kind. The moments and central moments are obtained. Then, the Voronovskaja-type asymptotic formula is investigated and point-wise estimates of these operators are studied. Also, some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K -functional. Finally, the rate of convergence and weighted approximation of these operators are presented.

Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

In this paper, we introduce a novel extension of the Bernstein-Kantorovich-Stancu type operator of degree n with the help of multiple shape parameters. Voronovskaja and Grüss-Voronovskaja type approximation theorems are examined via Ditzian-Totik moduli of smoothness. We investigate basic statistical convergence properties with respect to a non-negative regular summability matrix. Moreover, using Ditzian-Totik moduli, local and global approximation properties associated to the proposed operator have been established. Finally, several illustrative examples are presented to demonstrate the efficiency, applicability and validity of the operator. The graphical and numerical results verify that the proposed operator gives better approximation as well as expand the previous Bernstein-Kantorovich type modifications including single parameter.

A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

Residual-baseda posteriorierror analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier–Stokes and Darcy–Forchheimer equations, and derive, though in a non-standard sense, a reliable and efficient residual-baseda posteriorierror estimator suitable for an adaptive mesh-refinement method. For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Clément interpolant and Raviart–Thomas operator, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in localLpspaces, are the main tools for developing the efficiency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

ON (p, q)-STANCU-SZÁSZ-BETA OPERATORS AND THEIR APPROXIMATION PROPERTIES

In the present paper, we introduce the generalized form of (p; q)-analogue of the Szász-Beta operators with Stancu type parameters. We derived the local approximation properties of these operators and obtained the convergence rate and weighted approximation.