Duran, A. J.; Saff, E. B. Zero location for nonstandard orthogonal polynomials. (English) Zbl 1011.42013 J. Approximation Theory 113, No. 1, 127-141 (2001). The authors introduce an interesting method to derive bounds for the zeros of orthogonal polynomials with respect to an inner product of the form \[ \langle p,q\rangle = \sum_{k=0}^N \int p^{(k)}(z) {\overline q^{(k)}(z)}w_k(z) d\mu(z),\quad N\geq 1, \tag{a} \] where \(\mu\) is a finite positive Borel measure with support \(\Delta\subset {\mathbb C}\) (containing infinitely many points) and \(w_k\in L^1(\mu), w_k\geq 0, 0\leq k\leq N\). The main result is: 1. Assume \(w_k/w_{k-1}\in L^{\infty}(\mu)\) and write \(C_k=||w_k/w_{k-1}||_{\infty}\) \((1\leq k\leq N)\); if \(z_0\) is a zero of an orthogonal polynomial with respect to (a), then \[ d(z_0,\text{Co}(\Delta))\leq {1\over 2}\sqrt{\sum_{k=1}^N k^2C_k} \] (\(\text{Co}(\Delta)\) is the convex hull of \(\Delta\)). Moreover, the bound is sharp. 2. If, furthermore, \(0<\alpha_i, \beta_i< 2\) satisfy \(\alpha_i+\beta_i=2\) (\(1\leq i\leq N-1\)), \(\alpha_0=\beta_N=2\), then \[ d(z_0,\text{Co}(\Delta))\leq\max\left\{{k\sqrt{C_k}\over \sqrt{\alpha_{k-1}\beta_k}}, k=1,\ldots,N\right\}. \]The authors also study sequences of polynomials \(\{p_n\}_n\) satisfying a \((2N+3)\)-term recurrence relation \[ t^{N+1}p_n(t)=c_{n,0}p_n(t)+\sum_{k=1}^{N+1} \left[c_{k,k}p_{n-k}(t) + c_{n+k,k}p_{n+k}(t)\right], \] with initial conditions \(p_k=0\) \((k<0)\) and \(p_k(t)\) a polynomial of degree \(k\) for \(k=0,1,\ldots,N+1\). These are orthogonal with respect to \[ \langle p,q\rangle = \int (T_0(p),\ldots T_N(p))W(z)(T_0(q),\ldots,T_N(q))^{*}d\mu(z),\tag{b} \] where \[ T_m(p)=\sum_n {p^{(n(N+1)+m)}(0)\over (n(N+1)+m)!}t^n \] and \(\mu\) a positive measure supported on \({\mathbb R}\). Restricting themselves to the case \(W=\text{diag}(w_0,\ldots,w_N)\) and \(\mu\) a finite positive Borel measure with compact support in the complex plane, the authors then prove 3. Let the \(w_k\) be as before, \(|z|^2w_0(z)/w_N(z)\in L^{\infty}(\mu)\) and define \(C_0= |||z|^2w_0(z)/w_N(z)||_{\infty}\); if \(z_0\) is a zero of an orthogonal polynomial with respect to (b), then \[ |z_0|\leq\max\{\sqrt{C_0},\sqrt{C_1},\ldots,\sqrt{C_N}\}. . \] Reviewer: Marcel G.de Bruin (Delft) Cited in 7 Documents MSC: 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Keywords:zeros; higher-order recurrence relation; matrix polynomials; Sobolev inner product PDFBibTeX XMLCite \textit{A. J. Duran} and \textit{E. B. Saff}, J. Approx. Theory 113, No. 1, 127--141 (2001; Zbl 1011.42013) Full Text: DOI Link References: [1] Alfaro, M.; Marcellan, F.; Rezola, M. L.; Ronveaux, A., On orthogonal polynomials of Sobolev type: algebraic properties and zeros, SIAM J. Math. Anal., 23, 737-757 (1992) · Zbl 0764.33003 [2] Alfaro, M.; Marcellan, F.; Rezola, M. L.; Ronveaux, A., Sobolev type orthogonal polynomials: the non-diagonal case, J. Approx. Theory, 83, 266-287 (1995) · Zbl 0841.42013 [3] Duran, A. J., A generalization of Favard’s Theorem for polynomials satisfying a recurrence relation, J. Approx. Theory, 74, 83-109 (1993) · Zbl 0789.41017 [4] Duran, A. J., On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math., 47, 88-112 (1995) · Zbl 0832.42014 [5] Duran, A. J.; Van Assche, W., Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl., 219, 261-280 (1995) · Zbl 0827.15027 [6] Gautschi, W.; Kuijlaars, A. B.J., Zeros and critical points of Sobolev orthogonal polynomials, J. Approx. Theory, 91, 117-137 (1997) · Zbl 0897.42014 [7] Lopez Lagomasino, G.; Pijeira Cabrera, H., Zero location and \(n\)-th root asymptotics of Sobolev orthogonal polynomials, J. Approx. Theory, 99, 30-43 (1999) · Zbl 0949.42020 [8] G. Lopez Lagomasino, I. Perez Izquierdo, and, H. Pijeira Cabrera, Sobolev orthogonal polynomials in the complex plane, J. Comp. Appl. Math, in press.; G. Lopez Lagomasino, I. Perez Izquierdo, and, H. Pijeira Cabrera, Sobolev orthogonal polynomials in the complex plane, J. Comp. Appl. Math, in press. · Zbl 0973.42015 [9] Stahl, H.; Totik, V., General Orthogonal Polynomials (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0791.33009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.