GENERALIZED SIMULTANEOUS CHEBYSHEV APPROXIMATION

In this note,we develop,without assuming the Haar condition,a generalized simultaneousChebyshev approximation theory which is similar to the classical Chebyshev theory and con-rains it as a special case.Our results also contain those in[1]and[3]as a special case,and thetwo conjectures proposed by C.B.Dunham in[2]are proved to be true in the case of simulta-neous approximation.     

变阶可解族的带权逼近与插值逼近

本文讨论了变阶可解逼近族的插值逼近和带权逼近(权在插值点集Z上趋于无穷而在Z外为1)的关系.指出对变阶可解族而言,当逼近解为非亏损时,稠密性假设是自然满足的,且此时的最佳插值逼近等于该带权最佳逼近的极限.     

一类利用函数偶的联合最佳逼近

In this note, the author discusses the problem of best simultaneous approximation by function pairs. The existence and characterization theorems of the best simultaneous approximation in L₁ and L_∞ norms are obtained respectively, and some rezults of [ 1 ] are the special case of the theorems of the paper.     

VARIATIONAL L~q-APPROXIMATION BY VARISOLVENT FAMILY UNDER THE YOUNG'S CONDITION

1PreliminariesandExistenceForfEC[0,1],A.PinkusandO.Shi.h.[l]introducedthefollowingmeasures,'Thesupremaareattainable,andqE[1,co)Inthecaseq=1,wesimplyletWiththismeasurestA.PinkusandO.ShishahavestudiedbestapproximationbyalgebraicpolynondaISofdegree5n.Later,MaandShi[2]11avestudiedthecaseofbestapproximationfromratiollalpolyllollilalsR:,usingthesamegaugeill'JI1.TheyallgetthesimilarresultsforChebyshevapproximation,includingalternatiolltheoremascharacterisatiollforthebestappro~swhichisthegeom…     

Dunham型联合Chebyshev逼近理论

本文在不假定 G 在紧集 X 上满足 Haar 条件的情形,建立了 Dunham 型联合最佳逼近的 Chebyshev 理论,包括特征定理(定理5、定理6和推论2),唯一性与强唯一性定理(定理9和推论3)以及连续性定理,同时我们也得到了“削皮”定理与联合最佳逼近的量的对偶形式。(定理7)。     

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.