Korovkin-Type Approximation Theorem for Double Sequences of Positive Linear Operators via Statistical A-Summability

In this paper, using the concept of statistical A-summability which is stronger than the A-statistical convergence we provide a Korovkin-type approximation theorem on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. We also study the rates of statistical A-summability of positive linear operators.     

Korovkin-Type Approximation Theory in Riesz Spaces

In this paper, we prove the following Riesz spaces’ version of the Korovkin theorem. Let E and F be two Archimedean Riesz spaces with F uniformly complete, let W be a nonempty subset of \(E^{+}\), and let \((T_{n})\) be a given sequence of (r-u)-continuous elements of \(\mathcal {L(}E,F)\), such that \(\left| T_{n}-T_{m}\right| x=\left| (T_{n}-T_{m})x\right| \mathcal {\ }\)for all \(x\in E^{+},\) \(m,n\ge n_{0}\) (for a given \(n_{0}\in \mathbb {N} )\). If the sequence \((T_{n}x)_{n}\) \((r-u)\)-converges for every \(x\in W\), then \((T_{n})\) \((r-u)\)-converges also pointwise on the ideal \(E_{W}\), generated by W, to a linear operator \(S_{0}:E_{W}\rightarrow F\). We also prove a similar Korovkin-type theorem for nets of operators. Some applications for f-algebras and orthomorphisms are presented.     

关于一类复合幂级数展开式的收敛性定理

本文研究了一类复合型幂级数展开式,证明了一个收敛性定理并举例说明其应用.在注记中指出了可进一步研究的问题.     

Power Series of Bernstein Operators and Approximation of Resolvents

According to the theory developed by F. Altomare and his school, certain C ₀-semigroups can be approximated by iterates of positive linear operators. A. Albanese, M. Campiti and E. Mangino [J. Appl. Funct. Anal. 1 (2006), 343-358] proved that the resolvent (λ?A)^?1 of the infinitesimal generator of such a semigroup can be also approximated, for λ > 0, by suitable iterates. What happens when ${\lambda \to 0^{+}?}$ We give an answer in the case of the semigroup approximated by the classical Bernstein operators B _n on the canonical simplex S of ${\mathbb{R}^{d}}$ . Specifically, we show that $$-A^{-1}h = \lim\limits_{n \to \infty}\frac{1}{n}{\sum\limits^{\infty}_{k=0}}{B^{k}_{n}h}$$ for h in a certain subspace of C(S). This gives a new method to investigate the qualitative properties of the inverse of A.     

A Runge Approximation Theorem for Pseudo-Holomorphic Maps

The Runge approximation theorem for holomorphic maps is a fundamental result in complex analysis, and, consequently, many works have been devoted to extend it to other spaces (e.g. maps between certain algebraic varieties or complex manifolds). This article presents such a result for pseudo-holomorphic maps from a compact Riemann surface to a compact almost-complex manifold M, given that the manifold M admits many pseudo-holomorphic maps from ${\mathbb {C}{\rm P}^1}$ which can be thought of as local approximations of the Laurent expansion az +br ²/z. This result specializes to some compact algebraic varieties (e.g. rationally connected projective varieties). An application to Lefschetz fibrations is presented.     

A Strong Approximation Theorem for Quasi-associated Sequences

By combining the Csorgo-Révész quantile transform methods and the Skorohod-Strassen martingale embedding theorem, we prove a strong approximation theorem for quasi-associated random variables with mean zero and finite （2 ＋ δ）th moment under polynomial decay rate. As a consequence, the decay rate for a strong approximation theorem of associated sequences of Yu （1996） is weakened.     

A Strong Approximation Theorem for Stochastic Recursive Algorithms

The constant stepsize analog of Gelfand–Mitter type discrete-time stochastic recursive algorithms is shown to track an associated stochastic differential equation in the strong sense, i.e., with respect to an appropriate divergence measure.     

独立增量过程的一个强逼近定理

该文给出了独立增量过程的一种强逼近定理,由此得到相应的ｓｔｒａｓｓｅｎ重对数律．     

A Choquet-Deny-Type Theorem and Applications to Approximation in Weighted Spaces

We present a Choquet-Deny-type theorem in weighted spaces together with an application to simultaneous approximation and interpolation from inf-lattices generated by convex sets. Moreover, we determine a characterization of the Korovkin closure of vector lattices and, as a consequence, a different proof of the Stone-Weierstrass theorem for some weighted spaces.     

A Distortion Theorem for Univalent Gap Series

We prove a distortion theorem for conformal mappings of the unit disk for which log f is representable as the Hadamard gap series. This theorem implies in particular that such conformal mapping is almost bounded, i.e., for every >0, there is a positive constant C such that |f,(z)|C (1–|z|) ^–.     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

A Power Series Method for Treating Mixed Boundary Value Problems

A technique for treating certain mixed boundary value problemsis presented, and illustrated by its application to a probleminvolving short surface waves in infinitely long canals. Twocases are treated, namely canals of segmental and triangularcrosssections. The method consists of expanding a power series in a workingplane (a semi-circle), and results, in the examples given, inan infinite set of equations, although satisfactory numericalvalues may be obtained by truncation.     

A Joint Limit Theorem for General Dirichlet Series

We obtain a joint limit theorem in the sense of weak convergence of probability measures for general Dirichlet series in the space of meromorphic functions.     

The Weak Approximation Theorem for Valuations

(1) We prove a Weak Approximation Theorem for valuations that are not necessarily independent.(2) We study the existence of intersections of finite families of valuation rings having a prescribed divisibility group and prescribed residue fields.     

A Joint Limit Theorem for General Dirichlet Series

In this paper, a joint limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for general Dirichlet series is proved. The explicit form of the limit measure is given.     

关于weierstrass逼近定理的几点注记

Weierstrass逼近定理是函数逼近论中的重要定理之一,定理阐述了闭区间上的连续函数可以用一多项式去逼近.将该定理进行推广:即使一个函数是几乎处处连续的,也不一定具有与连续函数相类似的逼近性质,但是一个处处不连续的函数却有可能具有这样的性质.证明了定义在闭区间上且与连续函数几乎处处相等的函数具有类似的逼近性质,并给出了weierstrass逼近定理的一个推广应用.