M. Hosseini, J. J. Font
In this paper we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples.
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We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.
Quantitative estimates of convergence in nonlinear operator extensions of Korovkin's theorems
该研究针对实连续函数空间上的弱非线性单调算子序列，给出Korovkin定理非线性版本收敛性的连续模量化定量估计，并附示例阐释理论。
In this paper we prove analogues of Korovkin's theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the convergence almost everywhere, the convergence in measure and the convergence in $L^{p}$-norm. Several results illustrating the theory are also included.
The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type theorems are proved for convergence of completely positive maps with respect to weak, strong and uniform clustering of sequences of matrices of growing order. Such modes of convergence were originally considered for Toeplitz matrices (see [23],[26]). As an application, we translate the Korovkin-type approach used in the finite dimensional case, in the setting of …
Power series statistical convergence and an abstract Korovkin-type approximation theorem
本文通过证明$P$-统计收敛序列的子序列性质，在一般空间中建立抽象Korovkin型逼近定理，还研究了非正线性算子$r$阶推广及周期函数的相关逼近问题，推广改进了逼近论现有结果。
The convergence behavior of positive linear operators between certain locally convev vector lattices is reconsidered here.
On Korovkin-type theorems including exponential test functions on infinite intervals through power series convergence
本文针对经典收敛失效的情况，利用幂级数收敛性及积分可和性，建立了保留指数函数的Korovkin型逼近定理，并通过连续模分析了收敛速度。
Korovkin-type approximation results through summability methods in the Applied Science
本文通过可和性方法统一并推广了Korovkin型逼近定理，揭示了建立逼近结果所需的结构条件，并给出了正则矩阵收敛、理想收敛等新结果及后续研究方向。
In this paper we investigate some Korovkin type approximation properties of the q-Meyer-König and Zeller operators and Durrmeyer variant of the q-Meyer-König and Zeller operators via Abel summability method which is a sequence-to-function transformation and which extends the ordinary convergence. We show that the approximation results obtained in this paper are more general than some previous results. Finally, we obtain the rate of Abel convergence for the corresponding operators.
Operator version of Korovkin Theorem; Degree of Convergence and its Applications
本文针对Dumitru Popa提出的Korovkin定理算子版本建立定量估计，通过多种例子给出收敛度，并证明其三角类比，最终应用于Toeplitz结构大型线性系统的预处理问题。
Rough Weighted Ideal Convergence and Korovkin-Type Approximation via weighted equi-ideal convergence
本文引入粗糙加权理想收敛概念，通过加权等理想收敛提出Korovkin型逼近定理，扩展了等统计收敛并修正了相关定理。
Our main aim is to investigate the approximation properties for the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of approximation is determined. Also using weight function, the weighted statistical convergence theorem with the help of Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class is obtained. To support the convergence results of the proposed operators to the function, g…
Korovkin-type approximation for non-positive operators
本文发展几何方法，针对$C(X)$与$L^1[0,1]$上的一致有界算子序列证明Korovkin型定理，将其推广到$L^1(\boldsymbol{R})$情形，以非正的Grünwald插值算子为例验证该定理并给出数值示例。
The present work considers two important convergence techniques, namely deferred type statistical convergence and P-summability method in respect of positive linear operators. With regard to these techniques, we state and prove two general non-trivial Korovkin type approximation results for such operators. Further, we define an operator based on multivariate q-Lagrange-Hermite polynomials and exhibit the applicability of our general theorems.
In this paper, we introduce a modification of the Szasz-Mirakjan-Kantorovich operators as well as Stancu operators [9] (or a Dunkl generalization of modified Szasz-Mirakjan-Kantrovich operators [5]) which preserve the linear functions. These types of operators modification enables better error estimation than the classical Dunkl Szasz-MirakjanKantrovich as well as Stancu operators. We obtain some approximation results via well known Korovkin's type theorem, weighted Korovkin's type theorem convergence properties by using the modulus of continuity and the rate of convergence of the operators fo…
A Durrmeyer-type variant of Grünwald Interpolation Operators
本文构造了在L^p[0,π]空间上的格runwald插值算子的杜尔梅尔型变种，证明了其有界性和L^p范数下的收敛性，并利用Banach函数空间中的Korovkin型定理和模连续性得到收敛结果。
A Kantorovich-type variant of Grünwald Interpolation Operators
本文提出基于Chebyshev节点的Grünwald插值算子序列，扩展到L^p空间，并通过连续性和Kantorovich定理证明了在多种Banach函数空间中的收敛性。
Approximation by a new sequence of operators involving Laguerre polynomials
本文构造一类基于修正 Laguerre 多项式与 Păltănea 基的新积分算子，证明其在无界区间和加权空间中的收敛与逼近性质，并通过数值实验验证。
In this paper using the notion of an ideal I on a directed set, we extend the notion of convergence of nets of partial maps to the notions of I-convergence ( or filter convergence) of nets of partial maps and I*- convergence of nets of partial maps.
In this paper we consider the idea of I - convergence of nets of partial function from a metric space (X; d) to a metric space (Y; ?) and derive several basic characterization. This idea extends the concept of convergence of nets of partial function introduced by G. Beer et.al [1].
The Generalized Locally Toeplitz (GLT) sequences of matrices have been originated from the study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate some partial differential equations by discretization. The study of the asymptotic spectral behaviour of GLT sequence is very important in analysing the solution of corresponding partial differential equations. The approximating classes of sequences (a.c.s) and the spectral symbols are important notions in this connection. Recently, G. Barbarino obtained some additional results…
In a previous paper we considered a sequence of maps on a complete metric space $(X,d)$ and derived an extension of the Banach fixed point theorem. We showed that backward trajectories of maps $X\to X$ converge under mild conditions and that they can generate new types of attractors such as scale dependent fractals. Here we present two generalisations of this result and some potential applications. First, we study the structure of an infinite tree of maps $X\to X$ and discuss convergence to a unique "attractor" of the tree. We also consider "staircase" sequences of maps, that is, we consider a…
We prove an inverse function theorem of Nash-Moser type for maps between Fréchet spaces satisfying tame estimates. In contrast to earlier proofs, we do not use the Newton method, that is, we do not use quadratic convergence to overcome the lack of derivatives. In fact, our theorem holds when the map to be inverted is not C^2
We have derived that on certain Banach spaces having a graph structure $G$, the iterations for asymptotically $G$-nonexpansive map will converge weakly towards a fixed point. This result unifies and extends several theorems on fixed points proved by various authors for class of nonexpansive and asymptotically nonexpansive maps. As an application of this result, we derive that for maps satisfying the nonexpansive condition locally on special Banach spaces, the successive approximations converge weakly towards a fixed point.
Gromov-Hausdorff convergence of metric spaces of UCP maps
本文证明van Suijlekom条件技术保证UCP映射集合的Gromov-Hausdorff收敛，从而在谱信息不完整时仍能提取几何信息。
We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $β$, and then give convergence theorems for martingales of $β$-conditional expectations. We give the Birkhoff ergodic theorem for $β$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $β$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of…
This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $\ell^{1}$. We also point out some inaccurate assertions appearing in the literature on this topic.
We establish sharp estimates for the convergence rate of the Kranosel'ski\vı-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of Mathematics 199(2), 757-772, 2014) with constant $1/\sqrtπ$ is sharp and cannot be improved. To this end we consider the recursive bounds introduced in [3] (Proceedings of the 2nd International Conference on Fixed Point Theory and Applications, World Scientific Press, London, 27-66, 1992) which we reinterpret in terms of a nested family of optimal transport proble…
We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in the Hilbert's or Thompson's metric inherited from a convex cone. We show that the global uniqueness of the fixed point of the map, as well as the geometric convergence of every orbit to this fixed point, can be inferred from the semidifferential of the map at this point. In particular, we show that the geometric convergence rate of the orbits to the fixed point can be bounded in terms of Bonsall's nonlinear spectral radius of the semidiffer…
Coercivity and Gamma-convergence of the $p$-energy of sphere-valued Sobolev maps
研究球面值 Sobolev 映射在 $p\uparrow n$ 时的 $p$-能量，证明 Jacobi 矩阵收敛于积分 $m$-流且能量 Gamma-收敛为其质量，并推出极小映射与给定奇异性情形的新结果。
A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call {\sl pseudo-contractivity}, is introduced by means of the Abel averages of such maps. We show that the studied maps are dissipative in the spirit of the classical Lumer-Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.
Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence
本文刻画了从有限Toeplitz算子系统T_d到M_n的纯UCP映射，并证明其空间在矩阵Connes距离下Gromov-Hausdorff收敛到单位圆上归一化正矩阵值Borel测度空间。
Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that the iterates $f^k(x)$ converge to a fixed point for all $x \in C^\circ$. This condition generalizes the notion of type K order-preserving for maps on $\mathbb{R}^n_{>0}$. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogen…
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