Determination of Chebyshev approximations by nonlinear admissible subsets
Let B(X) denote the linear space of all bounded real-valued functions defined on a set X. In particular let \(w\in B(X)\) be such that \(\delta =\inf \{w(x): x\in X\}>0\). A weighted uniform norm \(\| \cdot \|_ w\) on B(X) is defined by \(\| f\|_ w=sp\{w(x)| f(x)|:\) \(x\in X\}\). Let G be a nonempty proper subset of B(X) and let f be a fixed element of B(X)\(\backslash G\). Denote \(\theta_ f=\inf \{\| f- h\|_ w:\quad h\in G\}.\) An element \(g\in G\) such that \(\theta_ f=\| f-g\|_ w\) is called the best weighted approximation to f in G. Let \(G_ f\) be the set of all such elements in G. Using the usual partial ordering of B(X) (\(f\leq g\) meaning f(x)\(\leq g(x)\) for all \(x\in X)\), the author specifies the notion of an ''admissible'' subset G of B(X) with respect to the pair \((f_ 1,f_ 2)\in B^ 2(X)\) and determines the set \(G_ f\) of all best weighted approximations to f by elements of an admissible subset G with respect to the pair \((f-\theta_ f/w,f+\theta_ f/w)\). In the sequel he applies his general theory if G is in particular: the set of semi-isotone functions, the set of functions whose modulus of continuity is bounded by a given modulus of continuity, the set of even functions. Some of these results develops the recent results of \textit{V. A. Ubhaya} [J. Approximation Theory 12, 146-159 (1974; Zbl 0288.41019), and ibid. 12, 315-331 (1974; Zbl 0292.41026); J. Math. Anal. Appl. 49, 659-679 (1975; Zbl 0299.41025)].
scientific article; zbMATH DE number 3844332