The theory of best approximation of functions
This article is an English translation of the article (in Russian) ``The Theory of best approximation of functions'' which appeared in 1945 in the collection'' `The Scientific Heritage of P. L. Chebyshev. I. Mathematics, pp. 122-172 (Nauka)'. This was the first major article that attempted to chronicle the beginning of approximation theory and is very useful for anyone who is interested in the history of approximation theory. The article is derived into following ten sections: 1. Approximating formulas by Poncelet 2. The memoir `The theory of mechanisms that are known under the name of parallelograms' (The works of P. L. Chebyshev, Mem. Acad. Sci. Petersb. vii (1854)) 3. The memoir `Some questions about minima connected to representing functions approximately' (the works of P. L. Chebyshev, Mem. Acad. Ser. Petersb. (6) vii (1859)) 4. Best approximation in a normed linear space 5. Necessary and sufficient conditions of approximation in Chebyshev problems 6. Least-squares approximation and the memoir `on functions that derivative least from zero' 7. Applications of the theory of best approximation by Chebyshev: Chebyshev's latest work 8. The extended problems solved by E. I. Zolotarev, A. A. Markov, V. A. Markov and N. I. Akhizer 9. The connection between best approximation and differential properties of a function. The work of S. N. Berstein 10. Further results dealing with best approximation of functions. A list of 18 references is given at the end.
scientific article; zbMATH DE number 1538243