We discuss Toeplitz operators on Fock-Sobolev space with positive measure symbols.By FockCarleson measure,we obtain the characterizations for boundedness and compactness of Toeplitz operators.We also give some equival...We discuss Toeplitz operators on Fock-Sobolev space with positive measure symbols.By FockCarleson measure,we obtain the characterizations for boundedness and compactness of Toeplitz operators.We also give some equivalent conditions of Schatten p-class properties of Toeplitz operators by Berezin transform.展开更多
The average σ-K width of the Sobolev-Wiener class in Lq(Rn) is studied for and the asymptotic behaviour of this quantity is determined. The exact value of average σ-K width of some class of smooth functions in L2(Rn...The average σ-K width of the Sobolev-Wiener class in Lq(Rn) is studied for and the asymptotic behaviour of this quantity is determined. The exact value of average σ-K width of some class of smooth functions in L2(Rn) is obtained.展开更多
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 ...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.展开更多
As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. In [a,b]. For a function f ∈...As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. In [a,b]. For a function f ∈ KWr[a,b], its values and derivatives up to r-1 order at a set of nodes x are known. These values are said to be the given Hermite information.This work reports the results on the best quadrature based on the given Hermite information for the class KWr[a,b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product,the best interpolation formula for the class KWr[a,b] is also obtained.展开更多
This paper concerns the problem of the Kolmogorov n-width, the linear re-width, the Gel'fand n-width and the Bernstein re-width of Sobolev classes of the periodic multivariate functions in the space Lp(Td) and the...This paper concerns the problem of the Kolmogorov n-width, the linear re-width, the Gel'fand n-width and the Bernstein re-width of Sobolev classes of the periodic multivariate functions in the space Lp(Td) and the average Bernstein o-width, average Kolmogorov o-widths, the average linear o-widths of Sobolev classes of the multivariate functions in the space LP(R ), where p = (p1,…,pd), 1 < Pj < ∞o, j = 1,2,…,d, or pj = ∞,j = 1,2,…, d. Their weak asymptotic behaviors are established for the corresponding quantities.展开更多
In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let Wp^r（B^d） be the usual Sobolev class of functions on the unit ball 54. We study the deviation fr...In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let Wp^r（B^d） be the usual Sobolev class of functions on the unit ball 54. We study the deviation from the radial function manifolds to WP^r（b^d）. Our results show that the upper and lower bounds of approximation by a linear combination of radial functions are asymptotically identical. We also find that the radial function manifolds and ridge function manifolds generated by a linear combination of ridge functions possess the same rate of approximation.展开更多
In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in ...In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.展开更多
This paper deals with the parallel information-based complexity of numerical integration on Sobolev class. We obtain tight bounds on the complexity, considered as a function of two variables simultaneously:the number... This paper deals with the parallel information-based complexity of numerical integration on Sobolev class. We obtain tight bounds on the complexity, considered as a function of two variables simultaneously:the number of processors, the rquired precision. This result seems to be new even in serial case.展开更多
1 Introduction Give a natural number r≥2. Let 1≤p, q≤∞. Put L<sub>pq</sub>（R）={f; ‖f‖<sub>pq</sub>【∞}, and W<sub>pq</sub><sup>r</sup>（R）={f∈L<sub>q<...1 Introduction Give a natural number r≥2. Let 1≤p, q≤∞. Put L<sub>pq</sub>（R）={f; ‖f‖<sub>pq</sub>【∞}, and W<sub>pq</sub><sup>r</sup>（R）={f∈L<sub>q</sub>（R）; f<sup>（r-1）</sup> locally absolutely continuous on the real axis R and ‖f<sup>（r）</sup>‖<sub>pq</sub>≤1}, where the norm ‖·‖<sub>pq</sub> is defined as follows （see [1]）: while Z=:{0, ±1, ±2,…}, and ‖·‖<sub>L<sub>p</sub></sub> denotes the usual L<sub>p</sub>-norm of an interval I. W<sub>pq</sub><sup>r</sup>（R） is called Sobolev-Wiener class. If p=q, W<sub>pp</sub><sup>r</sup>（R）= W<sub>q</sub><sup>r</sup>（R） is the usual Sobolev class. For convenience, we write W<sub>pq</sub><sup>r</sup>, L<sub>pq</sub> and ‖·‖<sub>p</sub> instead of W<sub>pq</sub><sup>r</sup> （R）, L<sub>pq</sub>（R） and ‖·‖<sub>pp</sub>,展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.11271092 and 11301101)Guangzhou Higher Education Science and Technology Project (Grant No.2012A018)
文摘We discuss Toeplitz operators on Fock-Sobolev space with positive measure symbols.By FockCarleson measure,we obtain the characterizations for boundedness and compactness of Toeplitz operators.We also give some equivalent conditions of Schatten p-class properties of Toeplitz operators by Berezin transform.
文摘The average σ-K width of the Sobolev-Wiener class in Lq(Rn) is studied for and the asymptotic behaviour of this quantity is determined. The exact value of average σ-K width of some class of smooth functions in L2(Rn) is obtained.
基金Supported by the National Natural Science Foundation of China (10971251, 11101220 and 11271199)the Program for new century excellent talents in University of China (NCET-10-0513)
文摘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.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10471128).
文摘As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. In [a,b]. For a function f ∈ KWr[a,b], its values and derivatives up to r-1 order at a set of nodes x are known. These values are said to be the given Hermite information.This work reports the results on the best quadrature based on the given Hermite information for the class KWr[a,b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product,the best interpolation formula for the class KWr[a,b] is also obtained.
基金The project is supported partly by the NationalNatural Science Foundation of China(10071007)and partly by the Foundation for University Key Teachers bythe Ministry of Education of China and partly by the Scientific Research Foundation for Returned Ov
文摘This paper concerns the problem of the Kolmogorov n-width, the linear re-width, the Gel'fand n-width and the Bernstein re-width of Sobolev classes of the periodic multivariate functions in the space Lp(Td) and the average Bernstein o-width, average Kolmogorov o-widths, the average linear o-widths of Sobolev classes of the multivariate functions in the space LP(R ), where p = (p1,…,pd), 1 < Pj < ∞o, j = 1,2,…,d, or pj = ∞,j = 1,2,…, d. Their weak asymptotic behaviors are established for the corresponding quantities.
基金supported by the National 973 Project (Grant No. 2007CB311002)National Natural Science Foundation of China (Grant Nos. 90818020,60873206)
文摘In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let Wp^r（B^d） be the usual Sobolev class of functions on the unit ball 54. We study the deviation from the radial function manifolds to WP^r（b^d）. Our results show that the upper and lower bounds of approximation by a linear combination of radial functions are asymptotically identical. We also find that the radial function manifolds and ridge function manifolds generated by a linear combination of ridge functions possess the same rate of approximation.
基金Supported by Project(No.10471010)of National Natural Science Foundation of ChinaSupported by the Development Foundation of Science and Technology of Tianjin Universities(20040405)Supported by Project"Representation Theory and Related Topics"of the"985 Program"of Beijing Normal University
文摘In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.
基金Supported by Institution of Higher Education Scientific Research Project in Ningxia(NGY2017011)Natural Science Foundations of Ningxia(NZ15055)+1 种基金Natural Science Foundations of China(1156105511461053)
基金this work was supported by china State Major Key Project for Basic Researchers
文摘 This paper deals with the parallel information-based complexity of numerical integration on Sobolev class. We obtain tight bounds on the complexity, considered as a function of two variables simultaneously:the number of processors, the rquired precision. This result seems to be new even in serial case.
基金Project supported by the National Natural Science Foundation of China.
文摘1 Introduction Give a natural number r≥2. Let 1≤p, q≤∞. Put L<sub>pq</sub>（R）={f; ‖f‖<sub>pq</sub>【∞}, and W<sub>pq</sub><sup>r</sup>（R）={f∈L<sub>q</sub>（R）; f<sup>（r-1）</sup> locally absolutely continuous on the real axis R and ‖f<sup>（r）</sup>‖<sub>pq</sub>≤1}, where the norm ‖·‖<sub>pq</sub> is defined as follows （see [1]）: while Z=:{0, ±1, ±2,…}, and ‖·‖<sub>L<sub>p</sub></sub> denotes the usual L<sub>p</sub>-norm of an interval I. W<sub>pq</sub><sup>r</sup>（R） is called Sobolev-Wiener class. If p=q, W<sub>pp</sub><sup>r</sup>（R）= W<sub>q</sub><sup>r</sup>（R） is the usual Sobolev class. For convenience, we write W<sub>pq</sub><sup>r</sup>, L<sub>pq</sub> and ‖·‖<sub>p</sub> instead of W<sub>pq</sub><sup>r</sup> （R）, L<sub>pq</sub>（R） and ‖·‖<sub>pp</sub>,
