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Excellent graduate-level monograph investigates relationship between various structural properties of real functions and the character of possible approximations to them by polynomials and other functions of simple construction. Based on classical approximation theorem of Weierstrass, P. L. Chebyshevs concept of the best approximation, converse theorem of S. N. Bernstein on existence of a function with a given sequence of best approximations. Each chapter includes problems and theorems supplementing main text. 1963 edition. Bibliography.