Zeros of Sobolev Orthogonal Polynomials of Gegenbauer Type

Author: Groenevelt W.G.M.

Source: Journal of Approximation Theory, Volume 114, Number 1, January 2002 , pp. 115-140(26)

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Abstract:

Let {S_n}_n denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product lang f, g rang _S= int f(x) g(x) d₀(x)+ int f(x) g(x) d₁(x),where >0 and {d₀, d₁} is a so-called symmetrically coherent pair, with d₀ or d₁ the classical Gegenbauer measure (x²-1) dx, >-1. If d₁ is the Gegenbauer measure, then S_n has n different, real zeros. If d₀ is the Gegenbauer measure, then S_n has at least n-2 different, real zeros. Under certain conditions S_n has complex zeros. Also the location of the zeros of S_n with respect to Gegenbauer polynomials, is studied. © 2002 Elsevier Science (USA)

Keywords: Sobolev orthogonal polynomials; symmetrically coherent pairs; zeros; Gegenbauer polynomials

Language: English

Document Type: Research article

Affiliations: Faculty of Information Technology and Systems, Department of Applied Mathematical Analysis, Delft University of Technology, GA Delft, 2600, The Netherlands

Publication date: 2002-01-01