DELFT UNIVERSITY OF TECHNOLOGYREPORT 94-57 Zeros of orthogonal polynomialsin a non-discrete Sobolev spaceM

Let fS n g denote a set of polynomials orthogonal with respect to the Sobolev inner product hf; gi = Z b a f(x)g(x)d 0 (x) + Z b a f 0 (x)g 0 (x)d 1 (x); where 0. If d 0 = d 1 is the Jacobian measure, then for n 2 and suuciently large, S n has n diierent real zeros interlacing with the zeros of P ;; n?1. This result can be generalized to a situation where d 0 and d 1 are not identical, but are socalled generalized coherent pairs. Moreover, for this situation, the polynomials fS n g satisfy a 5-term recurrence relation. On the other hand an example is given where d 0 = d 1 and S n has many complex zeros.