Yongping Liu

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

Abstract For two subsets W and V of a Banach space X , let K n ( W, V, X ) denote the relative Kolmogorov n -width of W relative to V defined by K n ( W , V , X ) = inf L n sup f ∈ W inf g ∈ V ∩ L n ∥ f - g ∥ x , where the infimum is taken over all n -dimensional linear subspaces L n of X . Let W 2 (Δ r ) denote the class of 2π-periodic functions f with d -variables satisfying ∫ [ - π , π ] d | Δ r f ( x ) | 2 d x ≤ 1 , while Δ r is the r -iterate of Laplace operator Δ. This article discusses the relative Kolmogorov n -width of W 2 (Δ r ) relative to W 2 (Δ r ) in L q ([−π, π] d ) (1 ≤ q ≤ ∞), and obtain its weak asymptotic result.

WIDTHS AND AVERAGE WIDTHS OF SOBOLEV CLASSES

Abstract This paper concerns the problem of the Kolmogorov n-width, the linear n-width, the Gel’fand n-width and the Bernstein n-width of Sobolev classes of the periodic multivariate functions in the space Lp(Td) and the average Bernstein σ-width, average Kolmogorov σ-widths, the average linear σ-widths of Sobolev classes of the multivariate functions in the space Lp(Rd), where P = ( p 1 , ⋯ , p d ) , 1 ≤ p j ∞ , j = 1 , 2 , ⋯ , d , or p j = ∞ , j = 1 , 2 , ⋯ , d . Their weak asymptotic behaviors are established for the corresponding quantities.

Average case tractability of a multivariate approximation problem

Abstract Many authors have studied exponentially-convergent tractability (EC-tractability) in the worst case setting. Here, we study EC-tractability in the average case setting. Our problem is multivariate approximation over the space of continuous real functions equipped with a zero-mean Gaussian measure whose covariance kernel is given as a Korobov kernel. We obtain necessary and sufficient conditions for certain kinds of tractability, includingEC-tractability.

A note on tractability of multivariate analytic problems

In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Wo?niakowski.

Optimal recovery of isotropic classes of twice-differentiable functions defined on d-dimensional Euclidean space☆

Abstract In the paper, inspired by the works of V. F. Babenko, S. V. Borodachov and D. S. Skorokhodov, we consider the problem of optimal recovery of isotropic classes of twice-differentiable multivariate functions defined on Euclidean space R d , and get some exact results. This problem is connected with the optimal covering of R d in discrete geometry. What Babenko et al. considered is the same kind of the optimal recovery problem of an isotropic class of twice-differentiable multivariate functions defined on a compact set of R d and an isotropic class of twice-differentiable periodic functions.

The best constants in the Wirtinger inequality

This study aimed to determine the best constants in the Wirtinger inequality ∥f∥2 ≤ Cr∥f(r)∥ 2, where f is defined on [0, 1] with f(0) = f′(0) = ⋯ = f(r−1)(0) = 0. First, we referred the computation of Cr to the maximal eigenvalue of an integral type operator Ar. Second, we proved that the computation of the eigenvalues of Ar is equivalent to the solution of a Strum–Liouville problem with some boundary conditions and hence we referred the computation of Cr to finding the minimal zero of a function with one variable. Third, by comparing with a result of Lifshits, Papageorgiou, Woźniakowski, we obtained that the strong asymptotic order of Cr: limr→∞Cr ⋅ r! = 1 2.

The sharp jackson inequality for L2-approximation on the periodic cylinder

Abstract We consider Jackson inequality in L 2 ( B d × T , W κ , μ B ( x ) ) , where the weight function W κ , μ B ( x ) is defined on the ball Bd and related to reflection group, and obtain the sharp Jackson inequality E n − 1 , m − 1 ( f ) 2 ≤ K n , m ( τ , r ) w r ( f , t ) 2 , τ ≥ 2 τ n , λ , where τ n , λ is the first positive zero of the Gegenbauer cosine polynomial C n λ ( cos θ ) ( n ∈ ℕ ) .

THE RESEARCH PROGRESS OF BNU GROUP ON RELATIVE WIDTHS

In this paper, we consider the relative n-widths of two kinds of periodic convolution classes, and , whose convolution kernels K and G are NCVD-kernel and B-kernel. The asymptotic estimations of and are obtained for p = 1 and ∞, 1 ≤ q ≤ ∞. We also defined a new concept of the average relative widths and obtained the exact results of the average relative widths of the classes of some smooth functions in L2(Rd).

OPTIMAL QUADRATURE OF THE SOBOLEV CLASS W1r(R) DEFINED ON WHOLE REAL AXIS

Abstract In this paper, we study the optimal quadrature problem with Hermite-Birkhoff type, on the Sobolev class W1r(R) defined on whole real axis, and we give an optimal algorithm and determite its optimal error.