Characterization of the local Lipschitz constant
Let \(X\) be a closed subset of \([a,b]\) with at least \(n+1\) points, and let \(C(X)\) denote the space of continuous real valued functions on \(X\) endowed with the uniform norm. Let \(H_ n\) denote a Haar set of dimension \(n\), and let the best approximation of \(f\in C(X)\) in \(H_ n\) be \(B_ n(f)\). The local Lipschitz constant, introduced by \textit{J. R. Angelos}, \textit{M. S. Henry}, \textit{E. H. Kaufmann, jun.}, and \textit{T. D. Lenker} [J. Approximation Theory 43, 53-63 (1985; Zbl 0552.41018)], is defined by \[ \lambda^ e_ n(f)= \lim_{\delta\to 0+} \lambda_ n(f,\delta), \] where \[ \lambda_ n(f,\delta)= \sup\bigl\{\| B_ n(f)- B_ n(g)\|/ \| f- g\|;\;0< \| f- g\|< \delta,\;g\in C(X)\bigr\}. \] In this paper, using polynomials introduced by \textit{A. V. Kolushov} [Math. Notes 29, 295-306 (1981; Zbl 0506.41023)], a characterization for the local Lipschitz constant for the best approximation from a Haar set is given. This characterization is than used to study the existence of a uniform local Lipschitz constant.
scientific article; zbMATH DE number 636068