Commentary on the article by V. L. Goncharov, ``The theory of the best approximation of functions'' (Q1587645)

Following the preface written by \textit{C. de Boor} and \textit{A. Pinkus} [J. Approximation Theory 106, No. 1, 1 (2000)] which introduces the issue of the J. Approximation Theory including the commentary, the editors forced an English translation of the historical article by \textit{V. L. Goncharov} [J. Approximation Theory 106, No. 1, 2-57 (2000; Zbl 0974.21023)] ``The theory of best approximation of functions'' which originally appeard in Russian as part of the collection ``The Scientific Heritage of P. L. Chebyshev'' [Nauka, 122-172 (1945)]. Since Goncharev wrote his review of Chebychev's work in 1945, parts of it would be presented differently from a modern point of view. The author provides the reader with such a view by commenting on parts of Goncharev's article. He picks out some of Chebyshev's ideas on best approximation, e.g., the famous Chebyshev's alternation theorem, and reformulates them using a modern interpretation especially in terms of convex analysis. He then turns back to exact expressions for elements of best approximation given by Chebyshev and Zolotarev and connects them to the above mentioned theorem.