Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Abstract

Let {Q ^(α,β) _n (x)} ^∞_n=0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$$

where λ>0 and d μ _α,β(x)=(x−a)(1−x)^α−1(1+x)^β−1 dx, d ν _α,β(x)=(1−x)^α(1+x)^β dx with a<−1, α,β>0. Their inner strong asymptotics on (−1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q ^(α,β) _n are obtained.