Signals are often of random character since they cannot bear any information if they are predictable for any time t, they are usually modelled as stationary random processes .On the other hand, because of the inertia of the measurement apparatus, measured sampled values obtained in practice may not be the precise value of the signal X(t) at time tk (k∈Z), but only local averages of X(t) near tk. In this paper, it is presented that a wide (or weak ) sense stationary stochastic process can be approximated by generalized sampling series with local average samples.
We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.
