Strong asymptotics on the support of the measure of orthogonality for polynomials orthogonal with respect to a discrete Sobolev inner product
The authors consider a Sobolev inner product of the following form \[ \langle h,g\rangle=\int h(x)g(x)d\mu(x)+\sum_{j=1}^m\sum_{i=0}^{N_j} h^{(i)}(c_j){\mathcal L}_{j,i} (g_i,c_j)\tag \(*\) \] with \(\mu\) a finite positive Borel measure whose support \(S_{\mu}\) on the real line contains an infinite set of points and \({\mathcal L}_{j,i}(g_i,c_j)\) is the evaluation of a linear differential operator \({\mathcal L}_{j,i}\) acting on \(g\) and \({\mathcal L}_{j,N_j}\not= 0\;(j=1,\ldots,m)\). Under certain conditions of regularity (in terms of non-singularity of matrices of coefficients of \({\mathcal L}_{j,i}\) in which zero rows and columns have been suppressed) and the well-known condition on the coefficients in the three term recurrence relation for the orthogonal polynomials (i.e., \(\alpha_n\rightarrow 1/2,\;\beta_n\rightarrow 0\)) with respect to \(d\mu(x)=w(x)dx\) with \(\int_{-1}^1 \log{w(x)}/\sqrt{1-x^2}dx<\infty\), the authors study asymptotics for the monic orthogonal polynomials with respect to \((*)\) \textbf{on} the interval of support of the measure. This paper supplements the result on strong asymptotics outside the support as given in \textit{G. López, F. Marcellán} and \textit{W. Van Assche} [Constr. Approx. 11, No. 1, 107--137 (1995; Zbl 0840.42017)] and \textit{F. Marcellán} and \textit{W. Van Assche} [J. Approx. Theory 72, No. 2, 193--209 (1993; Zbl 0771.42014)].
scientific article; zbMATH DE number 1067296