Approximation of Second-Order Moment Processes from Local Averages

In the literature, very few researchers considered approximating Brownian motion using Bernstein polynomials. Kowalski 1 is the first one who uses this method. In fact, if we restrict Brownian motion on 0, 1 , it is a real process with finite second order moment. In this paper, we will approximate all of the complex second order moment processes on a, b by Bernstein polynomials and other classical operators by 2 . Therefore the research obtained generalize that of 1 . On the other hand, it is well known that the sampling theorem is one of the most powerful tools in signal analysis. It says that to recover a function in certain function spaces, it suffices to know the values of the function on a sequence of points. Due to physical reasons, for example, the inertia of the measurement apparatus, the measured sampled values obtained in practice may not be values of f t precisely at times tk k ∈ Z , but only local average of f t near tk. Specifically, the measured sampled values are