Geodesic
Three Stages in the Development of Approximation Theory
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 7-19

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Some fundamental ideas and the results of approximation theory based on them are reviewed. The review covers the development of this branch of mathematics from its origins to the end of the 20th century. 
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V. M. Tikhomirov. Three Stages in the Development of Approximation Theory. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 7-19. http://geodesic.mathdoc.fr/item/TM_2022_319_a0/