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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.

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
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