Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials
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Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials Oksana Bihun1
· Kathy Driver2
Received: 14 May 2019 / Accepted: 9 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract n Let {xk,n−1 }n−1 k=1 and {xk,n }k=1 , n ∈ N, be two sets of real, distinct points satisfying the interlacing property xi,n < xi,n−1 < xi+1,n , i = 1, 2, . . . , n − 1. In [15], n−1 n Wendroff proved that if pn−1 (x) = (x − xk,n−1 ) and pn (x) = (x − xk,n ) , then k=1
k=1
pn−1 and pn can be embedded in a non-unique monic orthogonal sequence {pn }∞ n=0 . We investigate a question raised by Mourad Ismail as to the nature and properties of orthogonal sequences generated by applying Wendroff’s Theorem to the interlacing λ (x) and (x 2 − 1)C λ (x), where {C λ (x)}∞ is a sequence of monic zeros of Cn−1 k n−2 k=0 ultraspherical polynomials and −3/2 < λ < −1/2, λ = −1. We construct an algorithm for generating infinite monic orthogonal sequences {Dkλ (x)}∞ k=0 from the two λ λ λ λ 2 polynomials Dn (x) := (x − 1)Cn−2 (x) and Dn−1 (x) := Cn−1 (x), which is applicable for each pair of fixed parameters n, λ in the ranges n ∈ N, n ≥ 5 and λ > −3/2, λ (x) and λ = −1, 0, (2k − 1)/2, k = 0, 1, . . .. We plot and compare the zeros of Dm λ Cm (x) for selected choices of m ∈ N and a range of values of the parameters λ and λ (x) and C λ (x) approach are n. For −3/2 < λ < −1, the curves that the zeros of Dm m substantially different for large values of m. In contrast, when −1 < λ < −1/2, the two curves have a similar shape while the curves are almost identical for λ > −1/2. Keywords Ultraspherical polynomials · Wendroff’s Theorem · Interlacing of zeros · Quasi-orthogonal polynomials
Oksana Bihun
[email protected] Kathy Driver [email protected] 1
Department of Mathematics, University of Colorado, 1420 Austin Bluffs Pkwy, Colorado Springs, CO 80918, USA
2
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa
Numerical Algorithms
Mathematics Subject Classification (2010) Primary 33C50 · Secondary 42C05
1 Introduction The monic ultraspherical polynomial Cnλ (x) is defined by the three-term recurrence relation [9, eqn.(8.18)] λ λ (x) − bnλ Cn−2 (x), λ = 0, −1, . . . ; n = 1, 2, . . . , Cnλ (x) = xCn−1
(1)
where (n−1)(n−2+2λ) , λ = 0, −1, . . . ; n = 1, 2, . . . 4(n−2+λ)(n−1+λ) (2) is orthogonal on (−1, 1) with respect For each λ > − 12 , the sequence {Cnλ (x)}∞ n=0
λ C−1 (x) ≡ 0, C0λ (x) = 1, bnλ =
1
to the weight function (1 − x 2 )λ− 2 and for each n ∈ N, n ≥ 1, the zeros of Cnλ (x) λ (x) interlace with the are real, simple, symmetric, lie in (−1, 1) and the zeros of Cn−1 λ zeros of Cn (x), n ≥ 2, (see [14, Theorem 3.3.2]) namely, − 1 < x1,n < x1,n−1 < · · · < xn−1,n < xn−1,n−1 < xn,n < 1. {xi,n }ni=1
(3)
Cnλ (x)
where are the zeros of in increasing order. As λ decreases below −1/2, two (symmetric) zeros of Cnλ (x) leave the interval (−1, 1) through the endpoints −1 and 1 (see [6, p. 296]) and remain r
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