The generalization of some theorems of P. P. Korovkin on positive operators

We introduce a class of linear operators, containing the positive operators, in which P. P. Korovkin's theorem on the conditions and order of convergence of positive operators hold. We consider also the class of linear operators, convergent to the derivative, in which similar theorems hold.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 785–794, June, 1973.     

Tight global linear convergence rate bounds for Douglas–Rachford splitting

Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.     

New Korovkin Type Theorem for Non-Tensor Meyer–König and Zeller Operators

In this paper, we introduce a certain class of linear positive operators via a generating function, which includes the non-tensor MKZ operators and their non-trivial extension. In investigating the approximation properties, we prove a new Korovkin type approximation theorem by using appropriate test functions. We compute the rate of convergence of these operators by means of the modulus of continuity and the elements of modified Lipschitz class functions. Furthermore, we give functional partial differential equations for this class. Using the corresponding equations, we calculate the first few moments of the non-tensor MKZ operators and investigate their approximation properties. Finally, we state the multivariate versions of the results and obtain the convergence properties of the multivariate Meyer–König and Zeller operators.     

The new forms of Voronovskaya’s theorem in weighted spaces

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.     

Sobolev spaces, Lebesgue points and maximal functions

In this note, we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgue points with respect to convergence associated with such maximal operators.     

On the Generalized Szász–Mirakyan Operators

In this paper, we construct sequences of Szász–Mirakyan operators which are based on a function ρ. This function not only characterizes the operators but also characterizes the Korovkin set ${\left \{ 1,\rho ,\rho ^{2} \right \}}$ in a weighted function space. We give theorems about convergence of these operators to the identity operator on weighted spaces which are constructed using the function ρ and which are subspaces of the space of continuous functions on ${\mathbb{R} ^{+}}$ . We give quantitative type theorems in order to obtain the degree of weighted convergence with the help of a weighted modulus of continuity constructed using the function ρ. Further, we prove some shape-preserving properties of the operators such as the ρ-convexity and the monotonicity. Our results generalize the corresponding ones for the classical Szász operators.     

Convergence and Rate of Approximation in BVΦ for a Class of Integral Operators

We obtain estimates and convergence results with respect to ?-variation in spaces BV_Φ for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.     

Kernel based approximation in Sobolev spaces with radial basis functions

In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported radial kernels, and use them to approximate Sobolev space functions with optimal convergence order. These kernels can be simply constructed and readily applied to practical problems. The results show that the approximation power depends on the precision of the sampling instrument and the density of the available data.     

Banach空间中关于(H,η)-增生算子的一类新变分包含组问题

In this paper,we first introduce a new class of generalized accretive operators named(H,η)-accretive in Banach space.By studying the properties of(H,η)-accretive,we extend the concept of resolvent operators associated with m-accretive operators to the new(H,η)-accretive operators.In terms of the new resolvent operator technique,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.     

Piecewise linear least squares approximations of Frobenius-Perron operators

We propose a piecewise linear numerical method based on least squares approximations for computing stationary density functions of Frobenius-Perron operators associated with piecewise C² and stretching mappings of the unit interval. We prove the weak convergence of the method for a class of Frobenius-Perron operators, and the numerical results show that it is also norm convergent and has a better convergence rate than the piecewise linear Markov approximation method.     

On a Multivariable Class of Mittag-Leffler Type Functions

The present paper studies and investigates a class of Mittag-Leffler type multivariable functions. We derive the necessary convergence conditions and establish several properties associated with this class and those related with the corresponding class of fractional integral operators. New extensions of the introduced definitions and special cases of some of the results are also pointed out.     

Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

Higher Order Riesz Transforms in the Ultraspherical Setting as Principal Value Integral Operators

In this paper we represent the kth Riesz transform in the ultraspherical setting as a principal value integral operator for every \({k \in \mathbb N}\). We also measure the speed of convergence of the limit by proving L ^p -boundedness properties for the oscillation and variation operators associated with the corresponding truncated operators.     

Isoinertial family of operators and convergence of KdV cnoidal waves to solitons

In this paper we show the convergence of Korteweg-de Vries cnoidal waves to the limit soliton. It is proved that the convergence is uniform and in H²-norm, as the period of the solutions tends to infinity. Families of Hill operators are also studied. We obtain a condition under which families of operators are isoinertial. This condition is satisfied for classes of Hill operators that are obtained by linearization. Our application is to the family of linearized operators at the KdV cnoidal waves. It is proved that this family is isoinertial and also the value of the inertial index is calculated.     

Remarks on the Convergence of Pseudospectra

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.     

A family of projective splitting methods for the sum of two maximal monotone operators

A splitting method for two monotone operators A and B is an algorithm that attempts to converge to a zero of the sum A + B by solving a sequence of subproblems, each of which involves only the operator A, or only the operator B. Prior algorithms of this type can all in essence be categorized into three main classes, the Douglas/Peaceman-Rachford class, the forward-backward class, and the little-used double-backward class. Through a certain “extended” solution set in a product space, we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in Hilbert space. Our algorithms are essentially standard projection methods, using splitting decomposition to construct separators. We prove convergence through Fejér monotonicity techniques, but showing Fejér convergence of a different sequence to a different set than in earlier splitting methods. Our projective algorithms converge under more general conditions than prior splitting methods, allowing the proximal parameter to vary from iteration to iteration, and even from operator to operator, while retaining convergence for essentially arbitrary pairs of operators. The new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Unbounded order continuous operators on Riesz spaces

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of \(E^\sim \).     

King type modification of q-Bernstein-Schurer operators

Very recently the q-Bernstein-Schurer operators which reproduce only constant function were introduced and studied by C. V. Muraru (2011). Inspired by J. P. King, Positive linear operators which preserve x ² (2003), in this paper we modify q-Bernstein-Schurer operators to King type modification of q-Bernstein-Schurer operators, so that these operators reproduce constant as well as quadratic test functions x ² and study the approximation properties of these operators. We establish a convergence theorem of Korovkin type. We also get some estimations for the rate of convergence of these operators by using modulus of continuity. Furthermore, we give a Voronovskaja-type asymptotic formula for these operators.     

The spline collocation method for parabolic boundary integral equations on smooth curves

Summary. We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces. Received May 17, 2001 / Revised version received February 19, 2002 / Published online April 17, 2002     

Local approximation properties of certain class of linear positive operators via <Emphasis Type="Italic">I</Emphasis>—convergence

In this study, we obtain a local approximation theorems for a certain family of positive linear operators via I—convergence by using the first and the second modulus of continuities and the elements of Lipschitz class functions. We also give an example to show that the classical Korovkin Theory does not work but the theory works in I—convergence sense.