Algebraic Approximation: A Guide to Past and Current Solutions

Preface.- 1 Some Notes on Trigonometric Approximation.- Early years.- Direct and converse results: a motivation.- Some asymptotic results.- 2 The End 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 τ modulus of Ivanov.- Ditzian-Totik moduli.- Felten modulus.- 4 Exact Estimates and Asymptotics.- Asymptotics for Lip1(M,[−1,1].- Estimates for Wr.- Asymptotics for Cr,w[−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.