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Some properties of zeros of Sobolev-type orthogonal polynomials. (English) Zbl 0862.33005
The authors consider the monic polynomials $$Q_n (n=1,2, \dots)$$ which are orthogonal with respect to a certain inner product involving also a discrete part (Sobolev-type product) of the form (1) $$\langle f,g \rangle = \int_Ifgd \mu+ \sum^r_{i=0} M_if^{(i)} (c)g^{(i)} (c)$$, where $$\mu$$ stands for finite positive Borel measure supported on an interval $$I \subset\mathbb{R}$$ while $$c\notin {\overset\circ I}$$ (the interior of $$I)$$, $$r\geq 1$$, $$M_i\geq 0$$ $$(i=0, \dots, r-1)$$ and $$M_r>0$$, and derive some properties of their zeros. The results are given in the form of two main theorems. One of them states that $$Q_n$$ has at least $$(n- \overline n)$$ changes of sign in the interior of the convex hull of $$I$$, where $$\overline n$$ denotes the number of terms in the discrete part in (1) whose order of derivatives is less than $$\overline n$$. The other main result concerns the interlacing properties of the zeros and gives the conditions under which $$Q_n$$ and $$Q_{n+1}$$ have common zeros.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
##### Keywords:
Sobolev-type product
Full Text:
##### References:
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