Continuous Almost Best Approximations in C [0,1

Abstract

We consider the problem on almost best approximations in the space C[0, 1]. Let e > 0. Recall that the operator of an almost best approximation, or the e-projection of an element x ∈ C[0, 1] on a subset L ⊂ C[0, 1], is defined as the set-valued mapping x → PL,e(x) = {z ∈ L : ‖x− z‖ ρ(x, L) + e}, where ρ(x, L) = infy∈L ‖x − y‖ is the distance from x to L. For e = 0, the operator PL,e is the operator of metric projection. A mapping G : x → z ∈ PL,e is called an e-selector (of the almost best approximation operator). If G is continuous, then it is referred to as a continuous e-selector. In 1859, Chebyshev [1] noticed that the set Rm,n = { amxm+···+a0 bnxn+···+b0 , ai, bj ∈ R } of algebraic rational fractions is an existence and uniqueness set (nowadays, these sets are referred to as Chebyshev sets) in C[0, 1]. Maehly and Witzgall [3] showed that the metric projection on Rm,n is discontinuous. Nevertheless, Konyagin [2] proved that for each e > 0 there exists a continuous e-selector on Rm,n . In the following, we prove the existence of a continuous e-selector on L for the case in which L belongs to a certain class of sets, which, in particular, includes the set of algebraic rational fractions as well as the set of piecewise linear functions. Let n 2 and m 0. Consider a nonempty open set D ⊂ [0, 1]n−1 and a continuous injective mapping F : D × Rm+1 → C[0, 1]. Let L = ImF For arbitrary e > 0, we define a mapping Ye,L : C[0, 1]×D → 2Rm+1 by setting Ye,L(f, x1, . . . , xn−1) = {(y0, . . . , ym) : ‖F (x1, . . . , xn−1, y0, . . . , ym)− f‖ 0, f ∈ C[0, 1], and (x1, . . . , xn−1) ∈ D, the set Ye,L(f, x1, . . . , xn−1) is bounded and convex. (ii) Let f ∈ C[0, 1], e > 0, and k ∈ {0, . . . , n−2}. Then for any x1, . . . , xk ∈ R, the set { xk+1 ∈ R : there exist xk+2, . . . , xn−1 ∈ R such that (x1, . . . , xn−1) ∈ D and Ye,L(f, x1, . . . , xn−1) = ∅ } is connected in R. (The empty set is connected by definition.) (iii) The mapping Ye,L is continuous at all points where Ye,L(f, x1, . . . , xn−1) = ∅. (The target space is equipped with the Hausdorff metric.) Theorem 1. Let L ∈ K and e > 0. Then there exists a mapping G ∈ C(C[0, 1], L) (a continuous e-selector) such that