In this paper,the explicit estimates of central moments for one kind of exponential-type operators are derived.The estimates play an essential role in studying the explicit approximation properties of this family of operators.Using the proposed method,the results of Ditzian and Totik in 1987,Guo and Qi in 2007,and Mahmudov in 2010 can be improved respectively.
Let BΩp , 1 ≤ p < ∞, be the space of all bounded functions from Lp(R) which can be extended to entire functions of exponential type Ω. The uniform error bounds for truncated Whittaker-Kotelnikov-Shannon series based on local sampling are derived for functions f ∈ BΩp without decay assumption at infinity. Then the optimal bounds of the aliasing error and truncation error of Whittaker-Kotelnikov-Shannon expansion for non-bandlimited functions from Sobolev classes U(Wpr(R)) are determined up to a logarithmic factor.
Using a new reduction approach,we derive a lower bound of quantum complexity for the approximation of imbeddings from anisotropic Sobolev classes B(W r p([0,1] d)) to anisotropic Sobolev space W s q([0,1] d) for all 1 ≤ p,q ≤∞.When p ≥ q,we show this bound is optimal by deriving the matching upper bound.In this case,the quantum algorithms are not significantly better than the classical deterministic or randomized ones.We conjecture that the bound is also optimal for the case p < q.This conjecture was confirmed in the situation s = 0.
We study the approximation of functions from anisotropic Sobolev classes B(W_p^r([0,1]~d)) and H¨older-Nikolskii classes B(W_p^r([0,1]~d)) in the L q([0,1] d) norm with q ≤ p in the quantum model of computation.We determine the quantum query complexity of this problem up to logarithmic factors.It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.
