On the Best Approximation in Non-Archimedean Normed Linear Spaces

ＯｎｔｈｅＢｅｓｔＡｐｐｒｏｘｉｍａｔｉｏｎｉｎＮｏｎ－ＡｒｃｈｉｍｅｄｅａｎＮｏｒｍｅｄＬｉｎｅａｒＳｐａｃｅｓｏｎＷｅｉｄｕｎ（邱维敦）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＬｏｎｇｙｏｎＮｏｒｍａｌＣｏｌｌｅｇｅ，Ｆｕｊｉａｎ，３６...     

Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces

Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces. As applications, some new minimax inequalities, section theorem, best approximation theorem, existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.     

Best L 1 Approximation of Truncated Functions,Whitney-Type and Borh–Favard-Type Inequalities

We present upper and lower estimates for the best constant C _n such that, if \({f \in W^{n+1}_1}\) [?1, 1], then $$E_n(f)_1 \leqq C_n||\varphi^{n+1}f^{(n+1)}||_1,$$ where \({\varphi(x) =\sqrt{1 - x^2}}\) . For the proof we first find out the polynomials of the best L ¹[?1, 1] approximation of some truncated functions.     

On the Order of Best Approximation by Polynomials in H_q~p Spaces

ＯｎｔｈｅＯｒｄｅｒｏｆＢｅｓｔＡｐｐｒｏｘｉｍａｔｉｏｎｂｙＰｏｌｙｎｏｍｉａｌｓｉｎＨ■ＳｐａｃｅｓＸｉｎｇＦｕｃｈｏｎｇ（邢富冲）（ＣｅｎｔｒａｌＵｎｉｖｅｒｓｉｔｙｆｏｒＮａｔｉｏｎａｌｉｔｉｅｓ，Ｂｅｉｊｉｎｇ，１０００８１）Ｃｏｍｍｕｎｉ...     

Polynomial Ensembles and Pólya Frequency Functions

Journal of Theoretical Probability - We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of...     

On Copositive Approximation in Spaces of Continuous Functions Ⅱ:The Uniqueness of Best Copositive Approximation

This paper is part Ⅱ of "On Copositive Approximation in Spaces of Continuous Functions". In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C(Q), then for any admissible function f ∈C(Q)\M, the best copositive approximation to f from M is unique.     

Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L ^p -operator norm for all 1 ≤ p ≤ ∞. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattered data L ^p -Marcinkiewicz–Zygmund inequalities for these functions for all 1 ≤ p ≤ ∞. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).     

Functional Inequalities and Hamilton–Jacobi Equations in Geodesic Spaces

We study the connection between the p-Talagrand inequality and the q-logarithmic Sololev inequality for conjugate exponents p ≥ 2, q ≤ 2 in proper geodesic metric spaces. By means of a general Hamilton–Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton–Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the p-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group.     

Best Simultaneous Approximation in LΦ(I,X)

Let X be a Banach space and Φ be an Orlicz function.Denote by LΦ(I,X)the space of X-valued Φ-integrable functions on the unit interval I equipped with the Luxemburg norm.For f1,f2,...,fm ∈ LΦ(I,X),a distance formula disΦ(f1,f2,...,fm,LΦ(I,G))is presented,where G is a close subspace of X.Moreover,some existence and characterization results concerning the best simultaneous approximation of LΦ(I,G)in LΦ(I,X)are given.     

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Uniqueness of Best L Approximation For Continuous Functions

This paper deals with the problem of uniqueness of best L approximationfor continuous functions. In this paper we use a new method to establish akind of characterization theorems and a sufficient condition for unicity spacean n-dimensional subspace of C〔a,b〕, from which every function f in C〔a,b〕has a unique best L approximation.     

Jackson–Stechkin-Type Inequalities for the Approximation of Elements of Hilbert Spaces

Ukrainian Mathematical Journal - We introduce new characteristics for elements of Hilbert spaces, namely, their generalized moduli of continuity ωφ(x, Lp,V ([0, δ])) and obtain new...     

Existence and Uniquenss of Best Simultaneous Approximation

ＥｘｉｓｔｅｎｃｅａｎｄＵｎｉｑｕｅｎｓｓｏｆＢｅｓｔＳｉｍｕｌｔａｎｅｏｕｓＡｐｐｒｏｘｉｍａｔｉｏｎＬｉｕＲｕｉｚｈｅｎ(刘瑞珍)（ＨｅｂｅｉＮｏｒｍａｌＣｏｌｌｅｇｅ，Ｓｈｉｊａｚｈｕａｎｇ）Ｏｎｔｈｅｓｉｍｕｌｔａｎｅｏｕｓａｐｐｒｏｘｉｍａｔ...     

Weighted Polynomial Inequalities with Doubling and A ∞ Weights

We consider weighted inequalities such as Bernstein, Nikolskii, Remez, etc., inequalities under minimal assumptions on the weights. It turns out that in most cases this mimimal assumption is the doubling condition. Sometimes, however, as for the Remez and Nikolskii inequalities, one needs the slightly stronger A _∈ fty condition. We shall consider both the trigonometric and the algebraic cases. August 20, 1997. Date revised: April 19, 1998. Date accepted: May 26, 1998.     

On Polynomial Inequalities on Exponential Curves in ℂ n

We discuss Bernstein–Walsh type inequalities for holomorphic polynomials restricted to curves of the form $$\bigl(z,e^{P_{1}(z)},e^{P_{2}(z)},\ldots,e^{P_{d}(z)}\bigr)\in\mathbb{C}^{d+1},$$ where P ₁,P ₂,…,P _d are fixed polynomials on ? (such that the functions z and $e^{P_{k}(z)}$ are algebraically independent). The existence of such inequalities automatically implies the existence of associated Bernstein–Markov inequalities on the derivatives of polynomials restricted to the curve. The d=1 case has been much discussed in the recent literature. However, the d>1 case requires different techniques, and that is the subject of this work.     

Best Simultaneous Approximation and Its Applications in C ℝ(Q)

We develop a theory of best simultaneous approximation for closed convex sets in C _?(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max _q?Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C _?(Q) and give various characterizations of best simultaneous approximation.     

On Rellich’s Inequalities in Euclidean Spaces

On domains of Euclidean spaces we consider inequalities for test functions and their Laplacians. We describe a family of domains having vanishing Rellich constants. For the Euclidean space of dimension 4 we present a new version of the Rellich inequality. In addition, we prove new one-dimensional Rellich-type integral inequalities for linear combinations of test functions and their derivatives of orders one and two.