This book contains an exposition of several results related with direct and converse theorems in the theory of approximation by algebraic polynomials in a finite interval. In addition, some facts concerning trigonometric approximation that are necessary for motivation and comparisons are included. The selection of papers that are referenced and discussed document some trends in polynomial approximation from the 1950s to the present day.

Preface.- 1 Some Notes on Trigonometric Approximation.- Early years.- Direct and converse results: a motivation.- Some asymptotic results.- 2 TheEnd Points Effect.- Two different problems.- Nikolskii¿s discovery.- Problems connected with Nikolskii¿s result.- Timan-type estimates.- Estimates with higher-order moduli.- Gopengauz-Teliakovskii-type estimates.- Characterization of some classes of functions.- Simultaneous approximation.- Zamansky-type estimates.- Fuksman-Potapov solution to the second problem.- Integral metrics.- Lp, 0 < p < 1.- TheWhitney theorem.- Other classes of functions.- 3 Looking for New Moduli.- The works of Potapov.- Butzer and the method of Fourier transforms.- The Ï'' modulus of Ivanov.- Ditzian-Totik moduli.- Felten modulus.- 4 Exact Estimates and Asymptotics.- Asymptotics for Lip1(M,[a¿''1,1].- Estimates for Wr.- Asymptotics for Cr,w[a¿''1,1].- Estimates for integrable functions.- Pointwise asymptotics.- 5 Construction of Special Operators.- Estimates in norm.- Timan-type estimates.- Gopengauz estimates.- Bernstein interpolation process.- Integral operators.- Simultaneous approximation.- Estimation with constants.- The boolean sums approach.- Discrete operators.- Bibliography.- Index.