Trigonometrie Approximation with Exponential Error Orders. I. Construction of Asymptotically Optimal Processes; Generalized de la Vallée Poussin Sums.
A universal optimal in order approximation of a general functional in the space of continuous periodic functions is constructed and its fundamental properties and some generalizations are investigated. As an application the approximation of singular integrals is considered and illustrated by numerical results.
AMS classification: 41A36, 41A10, 41A25, 41Al7.
The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in whose tail function decreases as is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on and the minimax risk for that class is discussed.