Polynomial inequalities, functional spaces and best approximation on the real semiaxis with Laguerre weights
The paper under review is devoted to summarize results on polynomial \(L^p_{w_\alpha}\) approximation of functions on \([0,+\infty)\), when \(w_\alpha\) is the well known Lagerre weight, that is \(w_\alpha (x)=x^\alpha e^{-x}\). The article, which is divided into four sections, mainly refers to [\textit{M. C. De Bonis, G. Mastroianni} and \textit{M. Viggiano}, Proc. Alexits Memorial Conference, (1999; Zbl 1041.41004)]. The first section deals with some results on the system of Laguerre orthonormal polynomials \(p_m(w_\alpha)\) which are directly in connection with the subject. Techniques and results on polynomial inequalities which are due to G. Freud, P. Nevai, D. Lubinsky, H. N. Mhaskar, E. B. Saff and others, are included in Section 2. A major problem in polynomial approximation is the connection between the smoothness of a function and the convergence order of its best approximation. It is the reason for which Section 3 presents weighted \(K\)-functionals and moduli of smoothness to construct the Besov space \(B^p_{s,q}(w_\alpha)\), introduced by De Bonis, Mastroianni and Viggiano in 1999. Direct and converse results for the best polynomial approximation in \(L^p_{w_\alpha}\), proved in the last mentioned work, are discussed in Section 4.
scientific article; zbMATH DE number 1893815