Sheldon M.Eisenberg猜想及其有关问题

Sheldon M.Eisenberg猜想及其有关问题 SHELDON M. EISENBERG’S CONJECTURE AND RELATED PROBLEMS 吴杰 Wu jie first-author 本文解决了Sheldon M.Eisenberg在文中提出的一个猜想,证明了下列主要结果:设{a n （x）}∞n=0是一完全单调序列,若则有H n （e k ;x）=e k （x）,（k=0,1,2,……;x∈[0,1]）,a i （x）=x k ,（i=0,1,2,……;x∈[0,1]）. In this paper, we solve the conjecture proposed by sheldon M. Eisenberg in [1] and discuss some related problems. Definition 1 I.et {a n （x）} n ∞ =0 be a sequence of real-valued functions defined on [0,1]. Put △ n α （x）=α n （x）-α n+1 （x）, △ p α n （x）=△(△ P-1 α n （x）). If △ p α n （x）≥0 for all xe[0,1] and all integers n, p≥0, then the sequence{α n （x）} n ∞ =0 is called totally monotone. Definition 2 Let {α n （x）} n ∞ =0 be a sequence of real-vallud functions defined on [0,1] and f（x） be a real-valued function defined on [0,1] . We call and generalized Bernstein polynomial and generalized Bernstein-kantorovitch polyno-mial respectively. ln the sequel, let e k （x）=x k for k=0,1,2,…. Theorem 1 Let {α n （x）} n ∞ =0 be a totally monotone sequence on [0,1].lfthen α f （x）=x 4 , （i=0,1,…; xe[0,1]）. Theorem 2 lf {α n （x）} n ∞ =0 is a totally monotone sequence of continuous functions on [0,1],then a necessary and sufficient condition that lira H n * （f;x）=f（x） uniformly On[0,1], for each feCEO, 1] , is α i （x）=x t for i=0,1,2 and xe[0,1]. Theorem 3 Let {α,（x）} n ∞ =0 be a totally monotone sequence. If then α j （x）=x i , （i=0,1,…; xe[0,1]）. 中国科学院科学基金 1984-04-01 2021-04-01 4