Simultaneous approximation from convex sets

Uniqueness of simultaneous approximations in continuous function spaces

“…The study of the simultaneous approximation problem has a long history; see for example [1][2][3][4][5][6][7][8][9] and references therein. Such problems can be viewed as special cases of vector-valued approximation, and in the case when m < +∞ and λ i = 1 for each i = 1, .…”

Best simultaneous approximation to totally bounded sequences in Banach spaces

“…In the special case when m = 2, this problem of approximating simultaneously continuous functions on a finite closed interval was first studied by Dunham in [1], where results on characterization and uniqueness of the best simultaneous approximation were obtained, while characterization and uniqueness results for a class of problems involving L p norms were given in [4]. A general treatment of a class of problems for the case when m = 2, which includes these problems in [1,4] as special cases, was given in [2]. Extensions to the case when m = ∞ have been considered in [8] for some special infinite sequences in a real Banach space, and in [9] for the general infinite sequences in a (real or complex) Banach space.…”

“…The case of finitely many is also a special case of the vector-valued approximation studied by Pinkus [10]. Here we are particularly interested in the kind of the best simultaneous approximation problems studied in [1,2,4,8,9]. The general setting of this kind problem is as follows.…”

“…The problems of best simultaneous approximations to a set of functions has recently been a subject of intensive study, see for example [1,2,3,4,5] in the case of finite many of functions and [6,7,8,9] in the case of infinite, respectively. The case of finitely many is also a special case of the vector-valued approximation studied by Pinkus [10].…”

Best N-Simultaneous Approximation in Lp(μ,X)

“…The problem of the best -simultaneous approximation in ( , ) has been deeply and extensively studied; see, for example, [3][4][5][6][7][8][9][10][11]. When is a reflexive subspace of , it was proved in [3] that ( , ) is -simultaneously proximinal in ( , ), where is the Lebesgue measure on = [0, 1].…”