A family of operators with discontinuous ranges and approximation and restoration of continuous functions

A class of operators whose construction is similar to but simpler than the construction of Landau operators is the basis for the design of a modified family of operators with discontinuous ranges. This family has good approximation properties with respect to approximation and restoration problems for continuous functions.     

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.     

The modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves

In this article we present and analyze the modified quadrature method for strongly elliptic boundary integral equations of negative order on smooth closed curves. The method employs a modification that extracts the singularity appearing in the kernel or in some of its derivatives. Moreover, the composite trapezoidal rule is used for approximation of the integral. The modified quadrature method is proved to have the maximal rate O(h^−β+1) of convergence for operators of order β < 0 in general, and O(h^−β+2) for a large class of operators appearing in applications. Some numerical experiments confirming our theoretical results are also presented.     

Approximation of functions by a new construction of Bernstein‐Chlodowsky operators: Theory and applications

The main motivation of this paper is to provide a generalization of Bernstein‐Chlodowsky type operators which depend on function τ by means of two sequences of functions. The newly defined operators fix the test function set {1, τ, τ²} . Then we present the approximation properties of newly defined operators, such as weighted approximation, degree of approximation and Voronovskaya type theorems. Finally, we present a series of numerical examples demonstrating the effectiveness of this newly defined Bernstein‐Chlodowsky operators for computing function approximation.     

On the order of approximation for the rational interpolation to |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper.For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are xi=(i/n)r(i=1,2, … ,n;r＞0), it is proved that the exact order of approximation is O(1/n), O(1/nlogn) and O(1/nr), respectively, corresponding to 0＜r＜1, r=1 and r＞1.     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

This paper discusses generalized interpolating splines which determined by n order linear differential operators, and the best operators of interpolating approximation in W_2~m spaces, The explicit constructive method for the reproducing kernel in W_2~m space is presented, and proves the uniformity of spline interpolating operators and the best operators of interpolating approximation W_2~m space by reproducing kernel. The explicit expression of approximation error on a bounded ball in W_2~m space, and error estimation of spline operator of approximation are obtained.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

由线性微分算子确定的样条是连接多项式样条与希氏空间中抽象算子样条的重要环节,对微分算子样条的研究,既可从更高的观点揭示和概括多项式样条,又可启示我们去发现抽象算子样条的一些新的理论和应用． Ｇｒｅｅｎ函数是研究微分算子样条的重要工具 ［１］,但在微分算子插值样条的计算及将样条用于数值分析中,再生核方法起着更重要的作用．文献［２］［３］给出了与二阶线性微分算子插值样条有关的再生核解析表达式;由此得到了二阶微分算子插值样条与空间Ｗ_２～１［ａ,ｂ］中最佳插值逼近算子的一致性;而且还利用再生核给出了Ｈｉ…     

Almost sure convergence theorems in von Neumann algebras

The subadditive sequences of operators which belong to a von Neumann algebra with a faithful normal state and a given positive linear kernel are considered. We prove the almost sure convergence in Egorov’s sense for such sequences. Partially supported by the Ministry of Science and Ministry of Absorption. Partially sponsored by a grant from the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Korovkin-type convergence results for non-positive operators

Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. In this work we present qualitative Korovkin-type convergence results for a class of sequences of non-positive operators, more precisely regular operators with vanishing negative parts under a limiting process. Sequences of that type are called sequences of almost positive linear operators and have not been studied before in the context of Korovkin-type approximation theory. As an example we show that operators related to the multivariate scattered data interpolation technique moving least squares interpolation originally due to Lancaster and Šalkauskas [Surfaces generated by moving least squares methods, Math. Comp., 1981, 37, 141–158] give rise to such sequences. This work also generalizes Korovkin-type results regarding Shepard interpolation [Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38, 170–176] due to the author. Moreover, this work establishes connections and differences between the concepts of sequences of almost positive linear operators and sequences of quasi-positive or convexity-monotone linear operators introduced and studied by Campiti in [Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo (2) Suppl., 1993, 33, 229–238].     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.     

近似函数高阶导数的高精度积分逼近方法

主要研究稳定计算近似函数的高阶导数的积分逼近方法,方法因由Lanczos提出故也称为Lanczos算法.利用Legendre多项式的正交性,提出了一类逼近近似函数高阶导数的高精度积分方法,即构造出一系列积分算子Dn,h(m)去逼近噪声函数的高阶导数,且这些积分算子具有O(δ(2n+2)/(2n+m+2))的收敛速度,其中δ为近似函数的噪声水平.数值模拟结果表明提出的方法是稳定而有效的.     

Galerkin method for Wiener-Hopf operators with piecewise continuous symbol

This paper is concerned with the applicability of maximum defect polynomial (Galerkin) spline approximation methods with graded meshes to Wiener-Hopf operators with matrix-valued piecewise continuous generating function defined on R. For this, an algebra of sequences is introduced, which contains the approximating sequences we are interested in. There is a direct relationship between the stability of the approximation method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras are used in order to derive stability criteria for the approximation sequences.Supported by grant Praxis XXI/BD/4501/94 from FCT.Partly supported by FCT/BMFT grant 423.     

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.     

Equivalent theorems on simultaneous approximation by combinations of Gamma operators

In this paper,some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωr(4)λ(f,t)w(4)s(O≤λ≤1)are given.The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.     

Algebraic Properties of Subdivision Operators with Matrix Mask and Their Applications

Subdivision operators play an important role in wavelet analysis. This paper studies the algebraic properties of subdivision operators with matrix mask, especially their action on polynomial sequences and on some of their invariant subspaces. As an application, we characterize, under a mild condition, the approximation order provided by refinable vectors in terms of the eigenvalues and eigenvectors of polynomial sequences of the associated subdivision operator. Moreover, some necessary conditions, in terms of nondegeneracy and simplicity of eigenvalues of a matrix related to the subdivision operator for the refinable vector to be smooth are given. The main results are new even in the scalar case     

Convergence of distorted Brownian motions and singular Hamiltonians

We prove a convergence theorem for sequences of Diffusion Processes corresponding to Dirichlet Forms of the kind .We obtain convergence in total variation norm of the corresponding probability measures on the path space C(₊;^d) under hypotheses which, for example, are satisfied in the case of H _loc ¹ ( ^d )-convergence of the 's, but we can allow more singular situations as regards the approximating sequences. We use then these results to give a criterion of convergence for generalized Schrödinger operators in which the potential function should not necessarily exists as a measurable function. We obtain convergence not only in strong resolvent sense, but we also obtain convergence in the uniform operator topology up to sets of arbitrarily small Lebesgue measure. Applications to the problem of the approximation of ordinary Schrödinger operators by generalized ones corresponding to zero-range interactions are given.     

Some remarks on weakly singular integral operators

The eigenvalues _n of weakly singular integral operators, the order of singularity of the kernel of which is one half of the dimension of the domain, are shown to be of order O((1n(n+1)/n^1/2). There are convolution operators which demonstrate that this order cannot be improved in general.Research supported by the SFB 72 at the University of Bonn     

实轴上Hermite插值算子的收敛性

The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is： Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C（-∞,∞） satisfy ｜y｜= O（e^（1/2-δo）xk^2,） and ｜f（x）｜t= O（e^（1-εo ）x2^）. Then for any given point x ∈ R, we have limn→Hn,（f, x） = f（x）.     

The number of connected sparsely edged graphs. IV large nonseparable graphs

The number of nonseparable graphs on n labeled points and q lines is u(n, q). In the second paper of this series an exact formula for u(n, n + k) was found for general n and successive (small) k. The method would give an asymptotic approximation for fixed k as n → ∞. Here an asymptotic approximation to u(n, n + k) is found when k = O(n^1/2) and an approximation to logu(n, n + k) when k < (1 - ?)(1/3 n)1/2. The problem of finding an approximation to u(n, q) when (q - n)/n1/2 → + → and q/n - 1/2 logn - 1/2 log logn → - ∞ is open.