New Characterization for Nonlinear Weighted Best Simultaneous Approximation

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 then, such problems have been extended extensively, see, for example, 1–7 and references therein. In particular, characterization and uniqueness results were obtained in 7 for a wide class of best simultaneously approximating problem, which includes early results as special cases. The setting for the problems considered here is as follows. LetR∞ be a real linear space consisting of some sequences in the field of real numbers R and e δij ∈ R∞ for each i ∈ N, where δij 1 if j i, and 0 otherwise. We endow a norm ‖ · ‖A on R∞ such that the norm is monotonic; that is, for av ∈ R∞ and a real sequence bv , the condition that |bi| ≤ |ai| for each i 1, 2, . . . , implies that b1, b2, . . . ∈ R∞ and ‖ bv ‖A ≤ ‖ av ‖A. Let {λi} ∈ R∞ be a fixed sequence of positive numbers. Let X be a complex normed linear space with the norm ‖ · ‖ and G ⊂ X. The problem considered here is, for a sequence xi in X with λi‖xi‖ ∈ R∞, finding g0 ∈ G such that