Journal of Inequalities and Applications 
Volume 3 (1999), Issue 4, Pages 401-419
doi:10.1155/S1025583499000260

Estimates for polynomials orthogonal with respect to some Gegenbauer–Sobolev type inner product

Ana Foulquié moreno,1 Francisco Marcellán,1 and Boris P. Osilenker2

 1Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, C. Butarque, 15, Leganés-Madrid 28911, Spain
2Department of Mathematics, Moscow State Civil Engineering University, Moscow, Russia
Received 25 January 1998; Revised 18 September 1998

Abstract

In this paper we obtain some estimates in [−1,1] for orthogonal polynomials with respect to an inner product of Sobolev-type 〈f,g〉=∫−11fgdμ0+∫−11f′g′dμ1 where dμ0=Γ(2α+2)2(2α+1)Γ2(α+1)(1−x2)αdx+M[δ(x+1)+δ(x−1)]dμ1=N[δ(x+1)+δ(x−1)],  M,N≥0  and  α>−1 Finally, the asymptotic behavior of such polynomials in [−1,1] is analyzed.