Coherent pairs and zeros of Sobolev-type orthogonal polynomials
In the paper being reviewed the author deals, for fixed \(\lambda\geq 0\), with the set \(\{P^ \lambda_ n\}^ \infty_{n=0}\) of polynomials which are orthogonal with respect to a nondiscrete Sobolev inner product of the form \[ \langle f,g\rangle:=\int^ b_ af(x)g(x)d\psi_ 0(x)+\lambda\int^ b_ 0f'(x)g'(x)d\psi_ 1(x), \] where \(-\infty\leq a<b\leq\infty\) and \(\psi_ 0\), \(\psi_ 1:(a,b)\to\mathbb{R}\) are suitable distribution functions. Up to new results on the zeros of \(P^ \lambda_ n\) were only known for special choices of \(\psi_ 0\) and \(\psi_ 1\). For example, in the case \(d\psi_ 0(x)=d\psi_ 1(x)=dx\) it was proved that, for sufficiently large \(\lambda\), the zeros of \(P^ \lambda_ n\) are real and interlace with the zeros of the \((n-1)\)-th Legendre polynomial. But using the concept of coherent and symmetrically coherent pairs of measures, the author is able to derive more general results for the zero distribution of \(P^ \lambda_ n\). Specifically, he shows that in the coherent case, for \(n\geq 2\) and sufficiently large \(\lambda\), the zeros of \(P^ \lambda_ n\) are real, simple and interlace with the zeros of \(Q_{n-1}\) and \(R_{n-1}\), where \(\{Q_ n\}^ \infty_{n=0}\) and \(\{R_ n\}^ \infty_{n=0}\) denote sets of polynomials orthogonal with respect to \(d\psi_ 0\), respectively \(d\psi_ 1\).
scientific article; zbMATH DE number 469229