We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.

An estimation of approximation of continuous functions by positive linear operators in weighted norm using the weighted modulus of continuity is established. Application of the main result to the known Gadjiyev∗ -Ibragimov operators is given.

Let {Lt ; (x, t) ∈ R1 ×R1 +} denote the local time of Brownian motion and αt := ∫ ∞ −∞ (Lt ) 2 dx. Let η = N(0, 1) be independent of αt. For each fixed t ∫∞ −∞(L x+h t − Lt )2 dx− 4ht h3/2 L → ( 64 3 )1/2 √ αt η, as h → 0. Equivalently ∫∞ −∞(L x+1 t − Lt )2 dx− 4t t3/4 L → ( 64 3 )1/2 √ α1 η, as t → ∞.

In this paper, we obtain an upper bound of the modulus of continuity of linear multifractional multistable random motions. Such processes are generalizations of linear multifractional α-stable motions for which the stability index α is also allowed to vary in time. In the case of linear multifractional α-stable motions, we improve the recent result of [2]. The main idea is to consider some cond...

Let G be a bounded domain in the complex plane, let f be analytic in G and continuous in G, and let μ be a majorant, that is, a non-negative non-decreasing function defined for t ≥ 0 such that μ(2t) ≤ 2μ(t) for all t ≥ 0. Suppose that z1 ∈ ∂G and that |f(z1) − f(z2)| ≤ μ(|z1 − z2|) for all z2 ∈ ∂G. We show that then |f(z1)−f(z2)| ≤ Cμ(|z1−z2|) for all z2 ∈ G where C = 3456. If the assumption is...

We construct and analyze Bernstein-type operators on triangles with curved sides, their product and Boolean sum. We study the interpolation properties and approximation accuracy. Using the modulus of continuity we also study the remainders of the corresponding approximation formulas. Finally, there are given some particular cases and numerical examples.

We consider the class of integral operators Qφ on L (R+) of the form (Qφf)(x) = ∫ ∞ 0 φ(max{x, y})f(y)dy. We discuss necessary and sufficient conditions on φ to insure that Qφ is bounded, compact, or in the Schatten–von Neumann class Sp, 1 < p < ∞. We also give necessary and sufficient conditions for Qφ to be a finite rank operator. However, there is a kind of cut-off at p = 1, and for membersh...

*Correspondence: [email protected] 1Department of Mathematics and Computer Science, Wuyi University, Wuyishan, 354300, China Full list of author information is available at the end of the article Abstract In this paper, a kind of Schurer type q-Bernstein-Kantorovich operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of ...

In 1995, Yudovich extended his own 1963 uniqueness result for solutions to the 2D Euler equations with bounded initial vorticity to allow a certain class of initial vorticities whose L-norms grow no faster than roughly log p. Yudovich’s argument involves estimating part of the difference between two velocities in terms of the L∞-norm of each velocity. Because the two velocities have a (common) ...

This paper is concerned with functional variations in processes associatedwith Continuous Flow Models (CFMs). The considered CFMs are character-ized (defined) by their inflow-rate process, service-rate process, and buffer size,and have associated with them various derived processes such as the workload(buffer contents), outflow and overflow processes. The paper’s main result is<...