The zeros of linear combinations of orthogonal polynomials |
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| Authors: | A.F. Beardon K.A. Driver |
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| Affiliation: | aCentre for Mathematical Studies, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WB, UK;bThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Johannnesburg 2050, South Africa |
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| Abstract: | Let {pn} be a sequence of monic polynomials with pn of degree n, that are orthogonal with respect to a suitable Borel measure on the real line. Stieltjes showed that if m<n and x1,…,xn are the zeros of pn with x1<<xn then there are m distinct intervals f the form (xj,xj+1) each containing one zero of pm. Our main theorem proves a similar result with pm replaced by some linear combinations of p1,…,pm. The interlacing of the zeros of linear combinations of two and three adjacent orthogonal polynomials is also discussed. |
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| Keywords: | Orthogonal polynomials Linear combinations Interlacing zeros |
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