We study some quantitative estimates of the convergence of the iterates of some Rogosinski type operators to their associated cosine functions. We also consider a general cosine counterpart of the quantitative version of Trotter’s theorem on the approximation of C0-semigroups.

We introduce an index of convergence for double sequences of real numbers. This index is used to describe the behaviour of some bivariate interpolation sequences at points of discontinuity of the first kind. We consider in particular the case of bivariate Lagrange and Shepard operators.

Using a general procedure we consider some combination of different approximation processes by means of projections on orthogonal subspaces. We concentrate our attention on some particular positive approximation processes in spaces of square summable real functions in order to satisfy a prescribed Voronovskaja's type formula.

We discuss in more details the validity of a previous theorem and we give some clarifications in order to justify the validity of some previous results concerning the application of some general theorems to Bernstein operators.

It is proved the existence of a unique, global strong solution to the two-dimensional Navier-Stokes initial-value problem in exterior domains in the case where the velocity field tends, at large spatial distance, to a prescribed velocity field that is allowed to grow linearly.

Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out to be useful for a detailed description of some phenomena in interpolation theory at points of discontinuity of the first kind. In particular we give some applications to Lagrange operators on Chebyshev nodes of the first and second kind and Shepard operators.

We improve the quantitative estimate of the convergence in Trotter's approximation theorem and obtain a representation of the resolvent operators in terms of iterates of linear operators on its whole domain. We are able to apply these results in the very general context of Bernstein-Schnabl operators in an infinite dimensional setting. In the finite dimensional context of the standard simplex we give some estimates for functions of class C^{2,alpha} which considerably improve some previous estimates and allow to establish a partial inverse result.