Resumen de Liouville–weyl derivatives, best approximations, and moduli of smoothness
This work is devoted to the study of approximation and smoothness properties of the generalized Liouville-Weyl derivatives. The main three problems we address are the following.
First, we obtain new upper and lower estimates of norms and best approximations of the generalized Liouville-Weyl derivatives in terms of the best approximations of functions themselves. To deal with Liouville-Weyl derivatives, we make use of the concept of general monotonicity.
Second, we study a similar problem for moduli of smoothness of fractional order. More specifically, we prove new inequalities for moduli of smoothness of the generalized Liouville-Weyl derivatives via moduli of smoothness of functions themselves.
Both problems can be naturally divided into two cases: Lp-Lp and Lp-Lq estimates.
Third, we obtain new inequalities for best approximations by angle in the multivariate case.
