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.     

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.     

Existence and Uniquenss of Best Simultaneous Approximation

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

On the Best Degree of Approximation by Euler (E,q)-Means

Let f(x)∈L_(2π) and its Fourier series by f(x)～α_0/2+sum from n=1 to ∞(α_ncosnx+b_nsinx)≡sum from n=0 to ∞(A_n(x)). Denote by S_n (f,x) its partial sums and by E_n~q(f,x) its Euler (E, q)-means, i. e. E_n~q(f,x)=1/(1+q)~π sum from m=0 to n((?)q~(n-m)S_m(f,x)), with q≥0 (E_n~0≡S_n). In [1] Holland and Sahney proved the following theorem. THEOREM A Ifω(f,t) is the modulus of continuity of f∈C_(2π), then the degree of approximation of f by the (E,q)-means of f is givens by##特殊公式未编改     

On the Best Approximation in Non-Archimedean Normed Linear Spaces

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

从L~1(G,X)到L~p(G,X)的不变算子和乘子

设G是局部紧Abel群,X是半单纯的Banach代数。本文讨论了乘子M(L~1(G,X),L~p(G,X))和不变算子N(L~1(G,x),L~p(G,X)),证明了N(L~1(G,X),L~p(G,X))可与L(X,L~p(G,X))同一化,此处L(X,L~p(G,X))表示从X到L~p(G,X)的有界线性算子全体。得到了M(L~1(G,X),L~p(G,X))中元素T的表示,此外证明了M(L~1(G,X),L~P(G,X))≡N(L~1(G,X),L~p(G,x))的充要条件是dim X=1。     

On Converse Theorem of Best Approximation by Polynomials in Bergman Spaces H_q~p(p>0,q>1)

A Bernstein type theorem and a converse theorem of best approximation by polynomials inBergman spaces H_q~p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.     

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.     

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

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

Best Simultaneous Approximation for Dunham-Type by an N-Dimensional Subspace on a Set of N+1 Points

Let X={x_0,x_1…,x_n}and let c(X)be the set of all continuous real functions on X with the Chebyshev norm. Let G=span{g_1,g_2,…,g_n}be an n-dimensional subspace of c(X).Let T={(f~+,f~-):f~+≥f~-and f~+,f~-∈c(X)}.If there exists a P∈G such that max{||f~+-P||, ||f~--P||}=inf{max{||f~+-Q||, ||f~--Q||}:Q∈G},(1) then P is called a best simultaneous approximation to(f~+,f~-)from G.     

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.     

Pólya-Type Polynomial Inequalities in L P Spaces and Best Local Approximation

In this paper, we obtain an extension of the Pólya inequality for univariate real polynomials in L ^p spaces and new estimates for certain class of measurable sets. Inequalities for complex polynomials are also considered. We give an application to a multipoint best local approximation problem for real and complex polynomials.     

从L~p(G,A)到L~p(G,X)的L~1(G,A)-模同态和不变算子(1

于树模 《数学年刊A辑(中文版)》1990,(6)

Neue Lösungen der Funktionalgleichung für Matrizen Φ(X)Φ(Y)=Φ(XY)

Ohne Zusammenfassung     

Dual Density Conditions in (DF)— spaces,I

We define the two “dual density conditions” (DDC) and (SDDC) for locally convex topological vector spaces and study them in the setting of the class of (DF)- spaces (originally introduced by A. Grothendieck [14]). We show that for a (DF)- space E, (DDC) is equivalent to the metrizability of the bounded subsets of E, and prove that such a space E has (DDC) resp. (SDDC) if and only if the space l_∞(E) of all bounded sequences in E is quasibarrelled resp. bornological. As a consequence, we can then characterize the barrelled spaces \({\cal L}_b(\lambda_1,\ E)\) of continuous linear mappings from a Köthe echelon space λ₁ into a locally complete (DF)- space E; for purposes of a comparison, we also provide the corresponding characterization of the quasibarrelled resp. bornological (DF)- tensor products (λ₁)_b ^′ ?_ε E. Our results on the (DF)- spaces of type \({\cal L}_b(\lambda_1,\ E)\) and (λ₁)_b ^′) ?_ε E are of special interest in view of the recent negative solution, due to J. Taskinen (see [25]), of Grothendieck’s “problème des topologies” ([15]). — In part II of the article, we will treat weighted inductive limits of spaces of continuous functions and their projective hulls (cf. [6]) as an application. In his study of ultrapowers of locally convex spaces, S. Heinrich [16] had found it necessary to introduce the “density condition”. Our article [2] investigated this condition, mainly in the setting of Fréchet spaces, and with applications to distinguished echelon spaces λ₁. However, on the way to the main theorems of [2], it became apparent that the “right” setting for most of this material was a dual reformulation of the density condition in the context of (DF)- spaces, and this observation prompted the present research.     

ON ALMOST ISOMETRIES FROM L~1 (μ) INTO L~∞(v) OR C_b (△)

Let μ,ν be measures with diem L~1(μ) =dim L~∞(ν)=∞ and ⊿ be a "subinterval" of the real line. Let E=L~∞(ν) or C_b(⊿), in this paper it turns out that the IAP for the space B(L~1(μ)→E) has a negative answer.     

Approximation of Classes B p,θ r by Linear Methods and Best Approximations

We investigate problems related to the approximation by linear methods and the best approximations of the classes , 1 p in the space L .     

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.     

Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

We obtain asymptotic equalities for upper bounds of approximations of functions from the class C _β,∞^ψ by Poisson integrals in the metric of the space C.