Extension of Bernstein Polynomials to Infinite Dimensional Case

The purpose of this paper is to study some new concrete approximation processes for continuous vector-valued mappings defined on the infinite dimensional cube or on a subset of a real Hilbert space. In both cases these operators are modelled on classical Bernstein polynomials and represent a possible extension to an infinite dimensional setting. The same idea is generalized to obtain from a given approximation process for function defined on a real interval a new approximation process for vector-valued mappings defined on subsets of a real Hilbert space.