On Some Extremal Properties of Lagrange Interpolatory Polynomials

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show that

Full-size image

, where _n is a set of the linear operators with finite rank n+1 defined on −1,1], and where _n+1 denotes the set of polynomials p=∑_i=0ⁿ⁺¹a_ixⁱ of degreen+1 such that a_n+11. The infimum is achieved for Lagrange interpolatory polynomial for nodes .     

Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

函数类(?)_∞~r在Orlicz空间内的n宽度

讨论了由非退化的全正核K(x,y)和具有特定性质的函数系{ki(x)}ri=1所确定的某一函数类 (?)∞在Orlicz空间内的n-K宽度,n-G宽度,n-L宽度的精确估计问题.     

An expectation formula with applications

In this paper, we derive an expectation formula of a random variable having distribution W(x;q). As applications of the expectation formula, we give a transformation formula and an expansion of Sears? transformation formula.     

Condensed Cramer rule for some restricted quaternion linear equations

In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.     

The rotational dimension of a graph

Given a connected graph G = (N, E) with node weights s∈? and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of G: Find v_i∈?^|N|, i∈N so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter of all points is at the origin, and is maximized. In the case of a two‐dimensional optimal solution this corresponds to the equilibrium position of a quickly rotating net consisting of weighted mass points that are linked by massless cables of given lengths. We define the rotational dimension of G to be the minimal dimension k so that for all choices of lengths and weights an optimal solution can be found in ?^k and show that this is a minor monotone graph parameter. We give forbidden minor characterizations up to rotational dimension 2 and prove that the rotational dimension is always bounded above by the tree‐width of G plus one. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:283‐302, 2011     

Average widths of anisotropic Besov-Wiener classes

This paper concerns the problem of average σ-K width and average σ-L width of some anisotropic Besov-Wiener classes Srp q θb(Rd) and Srp q θB(Rd) in Lq(Rd) (1≤q≤p＜∞). The weak asymptotic behavior is established for the corresponding quantities.     

AVERAGE σ-K WIDTH OF CLASS OF Lp（R^n） IN Lq（R^n）

ＡＶＥＲＡＧＥ　σ－Ｋ　ＷＩＤＴＨ　ＯＦ　ＣＬＡＳＳ　ＯＦ　Ｌ_ｐ（Ｒ~ｎ）ＩＮ　Ｌ_ｑ（Ｒ~ｎ）ＡＶＥＲＡＧＥσ－ＫＷＩＤＴＨＯＦＣＬＡＳＳＯＦＬ_ｐ（Ｒ~ｎ）ＩＮＬ_ｑ（Ｒ~ｎ）￥ＬＩＵＹＯＮＧＰＩＮＧＡｂｓｔｒａｃｔ：Ｔｈｅａｖｅｒａｇｅσ－Ｋｗｉｄｔ...