Research in approximation theory in Russia dates back to P. L. Chebyshev's memoir ``The orie des me canismes connus sous le nom de paralle logrammes'' (Me m. Pre s. Acad. Imp. Sci. Pe tersb. Divers Savants, 1854, VII, 539 568). This memoir posed the problem of the best approximation of functions by polynomials and presented the first results concerning exact expressions for such approximations.
An interesting article by V. L. Goncharov, The theory of best approximation of functions,'' included in the collection Scientific Heritage of Chebyshev, Mathematics,'' Moscow, 1945, reviews Chebyshev's work and work of his collaborators in that early period when approximation theory was being established in Russia.
The present paper provides a brief commentary on the part of Goncharov's article devoted to the development of Chebyshev's ideas.
Goncharov justly points out that Chebyshev's memoir contains a series of mathematical facts and ... statements of utmost importance, undoubtedly forming the basis of his theory.'' Indeed, Chebyshev gives exact expressions for the best approximations to many different functions. The same subject is treated in a second memoir of Chebyshev, Sur les questions de minima qui se rattachent aÁ la repre sentation approximative des fonctions'' (Me m. Acad. Imp. Sci. Pe tersb. (6) Sci. Math. Phys. VII (1859), 199 291).
Let us consider both topics (foundation of the general theory and exact solutions) through the eyes of contemporary mathematicians.
1. STATEMENT OF PROBLEMS
In the first-mentioned memoir, Chebyshev wrote: ``Soit f (x) une fonction donne e, U un polynome du degre n avec des coefficients arbitraires. Si l'on choisit ces coefficients de manieÁ re aÁ ce que la diffe rence f (x)&U, depuis