The zeros of linear combinations of orthogonal polynomials

Authors:
A. F. Beardon;K. A. Driver
Affiliations:
Centre for Mathematical Studies, University of Cambridge, Cambridge, UK;The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannnesburg, South Africa
Venue:
Journal of Approximation Theory
Year:
2005

Citing 2
Cited 4

Common zeros of two polynomials in an orthogonal sequence

Journal of Approximation Theory
Quasi-orthogonality with applications to some families of classical orthogonal polynomials

Applied Numerical Mathematics

Interlacing of zeros of linear combinations of classical orthogonal polynomials from different sequences

Applied Numerical Mathematics
Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and Lq-minimal polynomials, q∈[1,∞]

Journal of Approximation Theory
A note on the interlacing of zeros and orthogonality

Journal of Approximation Theory
When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.00

Visualization

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.