Nonlinear weighted best simultaneous approximation in Banach spaces

Abstract

The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S-and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets and strict RS-sets. Our results obtained in the present paper extend and improve all earlier known results in this direction.

Best simultaneous approximation to totally bounded sequences in Banach spaces

“…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].…”

“…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.…”

“…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.…”

“…However, the study in [8,9] for the problem of best simultaneous approximations to infinite sequences is based on the following key assumption: Thus one interesting question arises naturally: can the assumption (1.2) be dropped in the study of simultaneous approximations to infinite sequences? This problem seems very difficult for the general case.…”

“…This problem seems very difficult for the general case. In fact, in the case when the assumption (1.2) is dropped, the method used in [8,9] does not work. In the present paper, we shall always assume that m = ∞ and develop a completely different technique to investigate the problem of best simultaneous approximations to totally bounded sequences in Banach spaces without assumption (1.2).…”