Direct and inverse theorems for Bernstein operators with inner singularities

We introduce a new type of Bernstein operators, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type of combinations are given.Comment: 13 pages, late

Approximation on Banach spaces of functions on the sphere

AbstractMany approximation results were proved on Lp(Sd-1), 1⩽p⩽∞ where Sd-1 is the unit sphere in Rd. We will show here that most of these results extend to Banach spaces on the sphere for which operation by a d×d orthogonal matrix is a continuous isometry

Approximation of *weak-to-norm continuous mappings

The purpose of this paper is to study the approximation of vector valued mappings defined on a subset of a normed space. We investigate Korovkin-type conditions under which a given sequence of linear operators becomes a so-called approximation process. First, we give a sufficient condition for this sequence to approximate the class of bounded, uniformly continuous functions. Then we present some sufficient and necessary conditions guaranteeing the approximation within the class of unbounded, *weak-to-norm continuous mappings. We also derive some estimates of the rate of convergence.Comment: 13 page

Jackson theorem in Lp,0<p<1, for functions on the sphere

AbstractThe best approximation of functions in Lp(Sd−1),0<p<1 by spherical harmonic polynomials is shown to be bounded by a modulus of smoothness recently introduced by the second author

Bernstein-type polynomials on several intervals

We construct the analogues of Bernstein polynomials on the set Js of s finitely many intervals. Two cases are considered: first when there are no restrictions on Js, and then when Js has a so-called T-polynomial. On such sets we define approximating operators resembling the classic Bernstein polynomials. Reproducing and interpolation properties as well as estimates for the rate of convergence are given. © Springer International Publishing AG 2017