Approximation of functions of two variables by certain linear positive operators

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an $r$th order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.Comment: 13 page

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussedComment: Research articl

Unbounded Functions and Positive Linear-Operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided

Simultaneous approximation by operators of exponential type

There are many results on the simultaneous approximation by sequences of special positive linear operators. In the year 1978, Ismail and May as well as Volkov independently studied operators of exponential type covering the most classical approximation operators. In this paper we study asymptotic properties of these class of operators. We prove that under certain conditions, asymptotic expansions for sequences of operators belonging to a slightly larger class of operators, can be differentiated term-by-term. This general theorem contains several results which were previously obtained by several authors for concrete operators. One corollary states, that the complete asymptotic expansion for the Bernstein polynomials can be differentiated term-by-term. This implies a well-known result on the Voronovskaja formula obtained by Floater

On Strong Approximation of Functions by Certain Linear Operators

This note is motivated by the results on the strong approximation of 2&#928;-periodic functions by means of trigonometric Fourier series.In this note is investigated certain class of positive linear operators in the polynomial weighted spaces. We introduce the strong differences of functions and their operators and we give the Jackson type theorems for them. We give also some corollaries.</p

Approximation numbers of composition operators on the H2H^2 space of Dirichlet series

By a theorem of Gordon and Hedenmalm, $\varphi$ generates a bounded composition operator on the Hilbert space $\mathscr{H}^2$ of Dirichlet series $\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and only if $\varphi(s)=c_0 s+\psi(s)$, where $c_0$ is a nonnegative integer and $\psi$ a Dirichlet series with the following mapping properties: $\psi$ maps the right half-plane into the half-plane $\operatorname{Re} s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\mathscr{H}^2$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when $c_0$ is positive. Both results are best possible. The case when $c_0=0$, $\psi$ is bounded and smooth up to the boundary of the right half-plane, and $\sup \operatorname{Re} \psi=1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $\varphi(s)=c_1+\sum_{j=1}^d c_{q_j} q_j^{-s}$ with $q_j$ independent integers, it is shown that the $n$th approximation number behaves as $n^{-(d-1)/2}$, possibly up to a factor $(\log n)^{(d-1)/2}$. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for $\mathscr{H}^2$ using estimates of solutions of the $\bar{\partial}$ equation. Finally, by a transference principle from $H^2$ of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.Comment: Final version, to appear in Journal of Functional Analysi

Analysis of Approximation by Linear Operators on Variable L

This paper is concerned with approximation on variable Lρp(·) spaces associated with a general exponent function p and a general bounded Borel measure ρ on an open subset Ω of Rd. We mainly consider approximation by Bernstein type linear operators. Under an assumption of log-Hölder continuity of the exponent function p, we verify a conjecture raised previously about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on the Lρp(·) space. Quantitative estimates for the approximation are provided for high orders of approximation by linear combinations of such positive linear operators. Motivating connections to classification and quantile regression problems in learning theory are also described