Sobolev-type orthogonal polynomials: The nondiagonal case
The authors study polynomials orthogonal with respect to an inner product of the form \[ (P, Q)= \int_I P(x) Q(x) d\mu+ \mathbb{P}(c)^t A\mathbb{Q}(c), \] where \(\mu\) is a positive Borel measure on the interval \(I\subset \mathbb{R}\), \(A\) is a \((r+ 1)\times (r+ 1)\) positive semidefinite matrix, \(c\in \mathbb{R}\), \[ \mathbb{P}(x):= \begin{pmatrix} P(x)\\ P'(x)\\ \vdots\\ P^{(r)}(x)\end{pmatrix} \] and \(\mathbb{P}(x)^t\) its transpose. They give representation formulas in terms of kernel matrices and derive some general results concerning the zeros of the Sobolev-type orthogonal polynomials. These zeros are studied in much more detail in the \((2\times 2)\)-case. Moreover, in this \((2\times 2)\)-case the relative asymptotic behaviour of the Sobolev-type orthogonal polynomials is studied in detail.
scientific article; zbMATH DE number 827937