The connection between systems of polynomials orthogonal with respect to different distribution functions

Uvarov, V.B.

doi:10.1016/0041-5553(69)90124-4

Cited by 95 publications

(106 citation statements)

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“…In the case of positive measures, there are many contributions (see [7][8][9]21]) when rational perturbationsω(x) = p(x) q(x) ω(x) of a weight function are considered. Grünbaum and Haine introduced a modification of such algorithms which allows to find the sequence of monic polynomials orthogonal with respect toL 3 in an alternative way to the approach by Maroni [17].…”

Section: Introductionmentioning

confidence: 99%

Darboux transformation and perturbation of linear functionals

Bueno¹

2004

Linear Algebra and its Applications

106

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Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and 1 x L + Cδ(x) where δ(x) denotes the linear functional (δ(x))(x k ) = δ k,0 , and δ k,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a three-term recurrence relation whose coefficients are the entries of the so-called monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix associated with the three canonical perturbations in terms of the monic Jacobi matrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobi matrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobi matrix associated with x 2 L as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals.

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Section: Introductionmentioning

confidence: 99%

Darboux transformation and perturbation of linear functionals

Bueno¹

2004

Linear Algebra and its Applications

106

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“…Unaware of the Jost-Kohn work and of each other, formulae for adding point masses for orthogonal polynomials on the real line case were found by Uvarov [20] and Nevai [11]. They found the perturbed polynomials, and Nevai computed the perturbed recursion coefficients.…”

Section: Resultsmentioning

confidence: 99%

Generalized Bounded Variation and Inserting Point Masses

Wong

2008

Constr Approx

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Abstract. Let dµ be a probability measure on the unit circle and dν be the measure formed by adding a pure point to dµ. We give a formula for the Verblunsky coefficients of dν following the method of Simon.Then we consider dµ 0 , a probability measure on the unit circle with ℓ 2 Verblunsky coefficients (α n (dµ 0 )) ∞ n=0 of bounded variation. We insert m pure points z j to dµ 0 , rescale, and form the probability measure dµ m . We use the formula above to prove that the Verblunsky coefficients of dµ m are in the form α n (dµ 0 ) + m j=1 zj n cj n + E n , where the c j 's are constants of norm 1 independent of the weights of the pure points and independent of n; the error term E n is in the order of o(1/n). Furthermore, we prove that dµ m is of (m + 1)-generalized bounded variation -a notion that we shall introduce in the paper. Then we use this fact to prove that lim n→∞ ϕ * n (z, dµ m ) is continuous and is equal to D(z, dµ m ) −1 away from the pure points.

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“…The above is a more efficient and generally applicable method of determining this family of orthogonal polynomials than that used in [4]. For other approaches to the problem of determining orthogonal polynomials when masses are added to measures under various conditions, see [10,11].…”

Section: Applicationsmentioning

confidence: 99%