In this paper,we study the optimal quadrature problem with Hermite-Birkhoff type,on the Sobolev class(R)defined on whole red axis,and we give an optimal algorithm and determite its optimal error.
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 opt...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.展开更多
文摘In this paper,we study the optimal quadrature problem with Hermite-Birkhoff type,on the Sobolev class(R)defined on whole red axis,and we give an optimal algorithm and determite its optimal error.
基金Supported by National Natural Science Foundation of China （Grant Nos. 10671019 and 10971251）, National Natural Science Special-purpose Foundation of China （Grant No. 10826079） and Chinese Universities Scientific Fund （Grant No. 2009-2-05）
文摘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.
