Zeros of Sobolev orthogonal polynomials of Hermite type
The authors study the polynomials \(\{S_n\}\) orthogonal with respect to a Sobolev inner product of the form \[ \int f(x)g(x) d\psi_0(x)+\lambda\int f'(x)g'(x) d\psi_1(x),\quad\lambda>0 \] with \(\{d\psi_0,d\psi_1\}\) a so-called symmetrically coherent pair of measures. In [J. Approximation Theory 89, 321-343 (1997; Zbl 0880.42012)] \textit{H. G. Meijer} showed that, for symmetrically coherent pairs, one of the two measures \(d\psi_0\) and \(d\psi_1\) must be a classical Hermite or Gegenbauer (ultraspherical) measure. In the paper under review the authors study the case of Hermite, which implies that \(\{d\psi_0,d\psi_1\}\) must be either \[ \left\{(x^2+\xi^2)e^{-x^2}dx,e^{-x^2} dx\right\}\quad\text{or}\quad \left\{e^{-x^2}dx,\frac{e^{-x^2}}{x^2+\xi^2} dx\right\}\quad\text{with}\quad\xi\neq 0. \] In each case the location of the zeros of \(S_n\) is determined compared with the zeros of the polynomials \(\{P_n\}\) orthogonal with respect to \(d\psi_0\) and those of the polynomials \(\{Q_n\}\) orthogonal with respect to \(d\psi_1\). In the first case it is shown that the polynomial \(S_n\) (of type I) has \(n\) different, real zeros and that the zeros of \(S_{n-1}\) separate those of \(S_n\). In the second case it is shown that the polynomial \(S_{2m+1}\) (of type II) has \(2m+1\) different, real zeros and that the polynomial \(S_{2m}\) has at least \(2m-2\) different, real zeros. Under certain conditions the polynomial \(S_{2m}\) might have two (complex conjugate) non-real zeros. In fact, it is proved that \(S_{2m}\) can only have two non-real zeros for \(\lambda\) sufficiently large and \(|\xi|\) sufficiently small.
scientific article; zbMATH DE number 1861202