Abstract
This paper is concerned with the problem of a wide class of weighted best simultaneous approximation in normed linear spaces, and it establishes a new characterization result for the class of approximation by virtue of the notion of simultaneous regular point.
Highlights
The problem of best simultaneous approximation has a long history and continues to generate much interest
Let R∞ be a real linear space consisting of some sequences in the field of real numbers R and ei δij ∈ R∞ for each i ∈ N, where δij 1 if j i, and 0 otherwise
We propose a same notion and the notion of simultaneous strongly regular point of a set for studying best simultaneous approximation to a sequence from the set in X, and establish new characterization results for this class of approximation problem
Summary
The problem of best simultaneous approximation has a long history and continues to generate much interest. The problem of approximating simultaneously two continuous functions on a finite closed interval was first studied by Dunham 1. Since such problems have been extended extensively, see, for example, 1–7 and references therein. We propose a same notion and the notion of simultaneous strongly regular point of a set for studying best simultaneous approximation to a sequence from the set in X, and establish new characterization results for this class of approximation problem. It should be remarked that our results are new even in the case when X is real noting that results obtained in this paper is valid for real normed linear spaces and when the approximated sequence is finite
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