Convergence of Certain Baskakov Operators of Integral Type

In the present paper, we propose a Baskakov operator of integral type using a function φ on [0,∞) with the properties: φ(0)=0,φ′>0 on [0,∞) and limx→∞φ(x)=∞. The proposed operators reproduce the function φ and constant functions. For the constructed operator, some approximation properties are studied. Voronovskaja asymptotic type formulas for the proposed operator and its derivative are also considered. In the last section, the interest is focused on weighted approximation properties, and a weighted convergence theorem of Korovkin’s type on unbounded intervals is obtained. The results can be extended on the interval (−∞,0] (the symmetric of the interval [0,∞) from the origin).

Estimation of Approximation with Jacobi Weights by Multivariate Baskakov Operator

We first give the unboundedness of multivariate Baskakov operators with the normal weighted norm. By introducing new norms, using the multivariate decomposition technique and the modulus of smoothness with Jacobi weight, the upper bound estimation of multivariate Baskakov operators is obtained. The obtained results not only generalize the corresponding ones for multivariate Baskakov operators without weights, but also give the approximation accuracy with the Jacobi weights approximation.

Direct and converse inequalities for positive linear operators on the positive semi-axis

We consider positive linear operators of probabilistic type L1f acting on real functions f defined on the positive semi-axis. We deal with the problem of uniform convergence of L1f to f, both in the usual sup-norm and in a uniform Lp type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of f. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.