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【期刊论文】Probabilistic and Average LinearWidths of Sobolev Space with Gaussian Measure in L∞-Norm
叶培新, Fang Gensun and Ye Peixin
Constr. Approx. (2004) 20: 159–172,-0001,():
-1年11月30日
In this paper we investigate the probabilistic linear (n, δ)-widths and paverage linear n-widths of the Sobolev space Wr 2 equipped with the Gaussian measure μ in the L∞-norm, and determine the asymptotic equalities λn,δ(Wr 2,μ, L∞) =开方( ln(n/δ)/ nr+(s−1)/2 , λ(a) n (Wr 2,μ, L∞)p=开方ln n/ nr+(s−1)/2 , 0 < p < ∞.
Probabilistic width, Average width, Gaussian measure, Sobolev space.,
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【期刊论文】Optimal Query Error of Quantum Approximation on Some Sobolev Classes
叶培新, SONG ZhanJie, , YE PeiXin
Science in China Series A: Mathematics 2008 Vol. 51 No. 1 1{16,-0001,():
-1年11月30日
We study the approximation of the imbedding of functions from anisotropic and generalized 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 LNp to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wrp ([0, 1]d)) to Lq([0, 1]d) space for all 1 6 q; p 6 1 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.
quantum approximation, Sobolev classes, n-th minimal query error
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叶培新, PeixinYe, Xiaofei Hu
Journal of Approximation Theory 150 (2008) 24 – 47,-0001,():
-1年11月30日
We study restricted Monte Carlo integration for anisotropic Hölder–Nikolskii classes. The results show that with clog2n random bits we have the same optimal order for the nth minimal Monte Carlo integration error as with arbitrary random numbers. We also study the computation of integration on anisotropic Sobolev classes in the quantum setting and present the optimal bound of nth minimal query error. The results show that the error bound of quantum algorithms is much smaller than that of deterministic and randomized algorithms.
Restricted Monte Carlo integration, Quantum integration, Anisotropic classes, Optimal error bound
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【期刊论文】Greedy approximation with regard to non-greedy bases
叶培新, V. N. Temlyakov, Mingrui Yang, Peixin Ye
,-0001,():
-1年11月30日
The main goal of this paper is to understand which properties of a basis are important for certain direct and inverse theorems in nonlinear approximation. We study greedy approximation with regard to bases with different properties. We consider bases that are tensor products of univariate greedy bases. Some results known for unconditional bases are extended to the case of quasi-greedy bases.
Greedy algorithm, m-term approximation, Greedy basis, Quasi-greedy basis
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【期刊论文】Cucker Smale Learning Theory in Besov Spaces
叶培新, Charles A. Micchelli, Yuesheng Xu and Peixin Ye
,-0001,():
-1年11月30日
Let B := (B, || · ||b) be a Banach space and H := (H, || · ||h) a dense subspace of B. We are interested in the rate at which the distance of a given x∈ B from the ball of radius t in H tends to zero as t →1. This problem has numerous applications and its recent importance in the Cucker Smale theory of learning renewed our interest in this subject. We study two aspects of this problem. First, we obtain general results on the relation between the CS-functional and the Peetre K-functional. Secondly, we estimate it in the concrete case of Besov spaces by using functions of exponential type.
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