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 B^pΩ, 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 ∈ B^pΩ 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 L/（Wp（R）） are determined up to a logarithmic factor.
Using a new reduction approach, we derive a lower bound of quantum com- plexity for the approximation of imbeddings from anisotropic Sobolev classes B （Wp^r （[0, 1]^d）） to anisotropic Sobolev space Wq^s（[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 al- gorithms 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.
For β 〉 0 and an integer r 〉 2, denote by H∞,β those 2π-periodic, real-valued functions f on R, which are analytic in Sβ = {z ∈ C： [ImzI 〈β} and satisfy the restriction If（r）（z）[ ≤ 1, z ∈ Sβ. The optimal quadrature formulae about information composed of the values of a function and its kth （k ： 1,..., r - 1） derivatives on free knots for the classes H∞,β are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.
We study the approximation of functions from anisotropic Sobolev classes b（WpR（[0, 1]d）） and HSlder-Nikolskii classes B（HPr（[0, 1]d）） in the Lq （[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.
We analyze the learning rates for the least square regression with data dependent hy- pothesis spaces and coefficient regularization algorithms based on general kernels. Under a very mild regularity condition on the regression function, we obtain a bound for the approximation error by esti- mating the corresponding K：-functional. Combining this estimate with the previous result of the sample error, we derive a dimensional free learning rate by the proper choice of the regularization parameter.
