A Korovkin-type theorem for sequences of positive linear operators on function spaces

Positivity - We prove a Korovkin-type theorem for sequences of positive linear operators on C([0, 1]). This theorem allows us to discuss some general results concerning positive linear...     

Korovkin-type theorem for sequences of operators preserving shape

In the paper we present Korovkin-type theorem concerning conditions of convergence sequences of linear operators preserving shape.     

<Emphasis Type="Italic">A</Emphasis>-summation process and Korovkin-type approximation theorem for double sequences of positive linear operators

The aim of this paper is to present a Korovkin-type approximation theorem on the space of all continuous real valued functions on any compact subset of the real two-dimensional space by using a A-summation process. We also study the rates of convergence of positive linear operators with the help of the modulus of continuity.     

Approximation by positive linear operators in modular spaces by power series method

In the present paper, we study the problem of approximation to a function by means of positive linear operators in modular spaces in the sense of power series method. Indeed, in order to get stronger results than the classical cases, we use the power series method which also includes both Abel and Borel methods. An application that satisfies our theorem is also provided.     

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from . The convergence features of these PCG have been naturally studied by means of the Weierstrass–Jackson Theorem [17,36,45], owing to the profound relationship between the spectral features of the matrices , generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f. On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Received October 1, 1996 / Revised version received May 7, 1998     

Interpolation theorem for linear operators

In generalizing a series of known results, the following theorem is proved: If K is a continuous linear operator mapping E₀ into F₀ and E₁ into F₁ (where E₀, E₁ and F₀, F₁, being ideal spaces, are Banach lattices of functions defined on ₁ and ₂ respectively), then for any . (0, 1) K maps E ₀ ^1– E ₁ into [(F ₀ )^1–(F ₁ )] and is continuous; for suitably chosen norms in the spacesE ₀ ^1– E ₁ and [(F ₀ )^1–(F ₁ )] the norm of K is a logarithmically convex function of . Six titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 593–598, December, 1967.The author wishes to thank M. A. Krasnosel'skii, under whose direction he is working.     

A Tauberian theorem for power series

In this paper we prove a complex Tauberian theorem for functions analytic in the open unit disc.The method used is based on Cauchy's integral formula for the Taylor coefficients. We show how a Blackwell type renewal theorem is an easy consequence of our main result.     

On Cartan's theorem for linear operators

In this paper, we describe a second main theorem of holomorphic curves in , of hyper‐order strictly less than 1, that involves a general linear operator . As an application, we derive a truncated second main theorem of degenerate holomorphic curves of hyper‐order strictly less than 1 using Nochka weights.     

A vanishing theorem for differential operators in positive characteristic

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A _n ) or a smooth quadric (in this case p is supposed to be odd) then Hⁱ( X,End( \sfF_*O_X ) ) = 0 {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \sfF_*O_X {\sf{F}_*}{\mathcal{O}_X} is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.     

A multivariate convergence theorem for positive sequences of functions

In this paper we explore the structure of certain generalized isometries of the special orthogonal group SO(n) which are transformations that leave any member of a large class of generalized distance measures invariant.     

A Seidel-Walsh theorem with linear differential operators

Assume that {S_n}₁^￥ {S_n}_1^infty is a sequence of automorphisms of the open unit disk Bbb D{Bbb D} and that {T_n}₁^￥{T_n}_1^infty is a sequence of linear differential operators with constant coefficients, both of them satisfying suitable conditions. We prove that for certain spaces X of holomorphic functions in the open unit disk, the set of functions f ? Xf in X such that {(T_n f) °S_n:  n ? Bbb N}{(T_n,f) circ S_n: , n in {Bbb N}} is dense in H(Bbb D)H({Bbb D}) is residual in X. This extends the Seidel-Walsh theorem together with some subsequent results.     

Weighted approximation of continuous functions by sequences of linear positive operators

In this work we obtain, under suitable conditions, theorems of Korovkin type for spaces with different weight, composed of continuous functions defined on unbounded regions. These results can be seen as an extension of theorems by Gadjiev in [4] and [5].