Convergence in Variation and Rates of Approximation for Bernstein-Type Polynomials and Singular Convolution Integrals

Convergence and rate of approximation in $BV^{\varphi}(\mathbb R^N_+)$ for a class of Mellin integral operators

“…We denote by Φ the class of all the functions ϕ : R From now on we will assume that ϕ ∈ Φ. We now recall some notations of the multidimensional setting in which we work (see, e.g., [14]). For f : R N + →R and x = (x 1 , .…”

“…Among them, there are the Mellin-GaussWeierstrass kernels (see [12] and also [9]), defined as [23,14,5]), and they are an example of kernels which fulfill all the previous assumptions. First of all they are of Fejér-type since P w (t) = w N P (t w ) with P (t) = Γ( and hence {P w } w>0 are an example of kernel functions to which our results can be applied.…”

“…Results about homothetictype operators in various settings can be found, for example, in [19,32,44,18,39,17,15,16,3,10,11,12], while for similar results about classical convolution operators see, e.g., [23,41,33,14,20,6,7,8,2,4].…”

“…This result extends to the multidimensional case an analogous one for the (one-dimensional) Musielak-Orlicz ϕ−variation ( [39]). In the case of the classical variation (see, e.g., [14] for translation operators) such result is an easy consequence of the integral representation of the variation for absolutely continuous functions; on the contrary, in the case of the ϕ−variation, due to the lack of an integral representation, it requires a more delicate direct construction.…”

Local approximation results for Szász-Mirakjan type operators

“…Szász-Mirakjan operators have an important role in the approximation theory, and their approximation properties have been investigated by many researchers (see, for instance, [1,2,3,5,7,11]). Recently, in [5] we have introduced the following modification of the classical Szász-Mirakjan operators…”