Direct and strong converse theorems for a general sequence of positive linear operators

Abstract

We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szász-Mirakyan operators.

Baskakov-Kantorovich operators reproducing affine functions: inverse results

“…Proposition 4. For each fixed q ∈ N, 1 ≤ q ≤ 6, there exists a constant C q such that, if λ > 1, x > 0 and λx ≥ 2 3 , then…”

Baskakov operators and Jacobi weights: pointwise estimates

“…where e 1 is the function defined by e 1 (t) = t. Proposition 3.2 For λ ≥ 1, one has (see [3]): (i) S λ,0 (x) = 1, S λ,1 (x) = 0, and for m ≥ 1,…”

“…The operators (7) have been studied by several authors (see [3,4,10,14], and [22]). Later we show that the operators W n,c (f , x) are related with Baskakov operators, but with a family of not integer parameters.…”

Approximation of functions and Mihesan operators

“…Problem.Can the rate of convergence to zero of B a n (f )−f be estimated as it was done in [3] or [5] for Baskakov operators? This problem will be study in a another paper.…”