A Note on the Chebyshev ε-Approximation Problem

This paper is concerned with the approximation of continuous functions in the Chebyshev norm. We let K be a compact topological space and we denote by C(K) the space of all real-valued continuous functions defined on K. We let f, f₁,…, f_n be linearly independent functions in C(K) and we denote by V the linear subspace spanned by f₁,…, f_n. Let ε ≥ 0. With Buck [2] and Singer [6], the following ε-approximation problem is considered: Find an element ḡ of V such that https://www.w3.org/1998/Math/MathML"> ∥ f − g ¯ ∥ ∞ ≤ ∥ f − g ∥ ∞ + ε  for all g ∈ V . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0192.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Here ‖ · ‖_∞ denotes the Chebyshev norm defined for each h ∈ C(K) by https://www.w3.org/1998/Math/MathML"> ∥ h ∥ ∞ = max k ∈ K | h ( k ) | . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003065098/b1b9c184-4054-4965-b4a1-a7c1424a86f6/content/eqn0193.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Such an element ḡ is called an ε-best approximation to f from V. It is highly desirable to have some characterization property which allows us to recognize an ε-best approximation.