Exact Markov inequalities for the Hermite and Laguerre weights

AbstractDenote by πn the set of all real algebraic polynomials of degree at most n and let Un≔{e-x2p(x):p∈πn}, Vn≔{e-xp(x):p∈πn}. We prove the following exact Markov inequalities:‖u(k)‖R⩽‖u*,n(k)‖R‖u‖R,∀u∈Un,∀k∈N,and‖v(k)‖R+⩽‖v*,n(k)‖R+‖v‖R+,∀u∈Vn,∀k∈N,where ‖·‖R (‖·‖R+) is the supremum norm on R (R+≔[0,∞)) and u*,n (v*,n) is the Chebyshev polynomial from Un (Vn)

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Weighted Sobolev spaces: Markov-type inequalities and duality

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities.Francisco Marcellán: Supported in part by two Grants from Ministerio de Economía y Competitividad (MTM2012-36732-C03-01 and MTM2015-65888-C4-2-P), Spain. Yamilet Quintana: Supported in part by a Grant from Ministerio de Economía y Competitividad (MTM2012-36732-C03-01), Spain. José M. Rodríguez: Supported in part by three Grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigacin (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México

Averaged cubature schemes on the real positive semiaxis

Stratified cubature rules are proposed to approximate double integrals defined on the real positive semiaxis. In particular, anti-Gauss cubature formulae are introduced and averaged cubature schemes are developed. Some of their appropriate modifications are also studied. Several numerical experiments are given to testify the performance of all the formulae