Orthogonal rational functions with complex poles: The Favard theorem

Deckers, Karl; Bultheel, Adhemar

doi:10.1016/j.jmaa.2009.03.064

Cited by 4 publications

(10 citation statements)

References 6 publications

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“…In the opposite direction as in Theorem 2.1, the following Favard-type theorem has been proved in [3]. Theorem 2.2.…”

Section: Preliminariesmentioning

confidence: 95%

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The existence and construction of rational Gauss-type quadrature rules

Deckers

Bultheel

2012

Applied Mathematics and Computation

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Abstract. Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f }. These are quadrature formulas with n positive weights and with the n−m remaining nodes real and distinct, so that the quadrature is exact in a (2n−m)-dimensional space of rational functions.

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“…In the opposite direction as in Theorem 2.1, the following Favard-type theorem has been proved in [3]. Theorem 2.2.…”

Section: Preliminariesmentioning

confidence: 95%

“…Thus, in what follows we can restrict ourselves to the case in which α n−1 ∈ R 0 and α n ∈ C {0,αn−1} . 3 We then have the following lemma.…”

Section: Preliminariesmentioning

confidence: 99%

The existence and construction of rational Gauss-type quadrature rules

Deckers

Bultheel

2012

Applied Mathematics and Computation

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“…In the opposite direction as in Theorem 1, the following Favard-type theorem has been proved in [14].…”

Section: Preliminariesmentioning

confidence: 97%

A generalized eigenvalue problem for quasi-orthogonal rational functions

2011

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In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among {α 1 , . . . , α n } ⊂ (C 0 ∪ {∞}), are not all real (unless α n is real), and hence, they are not suitable to construct a rational Gaussian quadrature rule (RGQ). For this reason, the zeros of a so-called quasi-ORF or a so-called para-ORF are used instead. These zeros depend on one single parameter τ ∈ (C ∪ {∞}), which can always be chosen in such a way that the zeros are all real and simple. In this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi-ORF and the corresponding weights in the RGQ. First, we study the connection between quasi-ORFs, para-ORFs and ORFs. Next, a condition is given for the parameter τ so that the zeros are all real and simple. Finally, some illustrative and numerical examples are given. Mathematics Subject Classification (2000)42C05 · 65D32 · 65F15

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“…respectively. Thus, it remains to prove that the rational functions φ n , ψ n ∈ L n \ {0} in (12) are unique up to a common non-zero multiplicative factor, as well as the fact that g n (β n ) ̸ = 0. We will prove both things simultaneously by induction on n.…”

Section: Orthogonal Rational Functions and Functions Of The Second Kindmentioning

confidence: 99%

“…Next, suppose that for 0 ⩽ k < n the rational functions φ k and ψ k in (12) are unique up to a non-zero multiplicative factor, and that g k (β k ) ̸ = 0. We then continue by induction to prove that the same holds true for k = n. Letφ n ,ψ n ∈ L n \{0}, thenφ n = k n φ n +a n−1 andψ n = k n ψ n +b n , with k n ∈ C, a n−1 ∈ L n−1 and b n ∈ L n .…”

Section: Orthogonal Rational Functions and Functions Of The Second Kindmentioning

confidence: 99%

An extension of the associated rational functions on the unit circle

Deckers

Cantero

Ituarte

et al. 2011

Journal of Approximation Theory

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A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients.The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Carathéodory function. Such representations are used to find new properties of the associated rational functions.

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Orthogonal rational functions with complex poles: The Favard theorem

Cited by 4 publications

References 6 publications

The existence and construction of rational Gauss-type quadrature rules

The existence and construction of rational Gauss-type quadrature rules

A generalized eigenvalue problem for quasi-orthogonal rational functions

An extension of the associated rational functions on the unit circle

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