On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

Ruiz, Herbert Dueñas; Marcellán, Francisco; Molano, Alejandro

doi:10.15446/recolma.v53n2.85524

Cited by 5 publications

(2 citation statements)

References 12 publications

(25 reference statements)

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“…Step k. For k ≥ 3, using the step 0 and the information in steps 1 to k − 1 to compute: ε Remark 5.1. Notice that an algorithm for generate Fourier-Sobolev coefficients when you deal with (1, 1) coherent pairs of measures with respect to the derivative operator has been presented in [10].…”

Section: Algorithm 1 Even Order Fourier Dunkl-sobolev Coefficientsmentioning

confidence: 99%

Some results on Dunkl-coherent Pairs

Sghaier

Hamdi

2020

Integral Transforms and Special Functions

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Let T µ be the Dunkl operator. A pair of symmetric measures (u, v) supported on a symmetric subset of the real line is said to be a symmetric Dunkl-coherent pair if the corresponding sequences of monic orthogonal polynomials {P n } n≥0 and {R n } n≥0 (resp.) satisfywhere {σ n } n≥1 is a sequence of non-zero complex numbers andIn this contribution we focus the attention on the sequence {S (λ,µ) n } n≥0 of monic orthogonal polynomials with respect to the Dunkl-Sobolev inner product < p, q > s,µ =< u, pq > +λ < v, T µ pT µ q >, λ > 0, p, q ∈ P.An algorithm is stated to compute the coefficients of the Fourier-Sobolev type expansions with respect to < ., . > s,µ for suitable smooth functions f such that fFinally, two illustrative numerical examples are presented.

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Section: Algorithm 1 Even Order Fourier Dunkl-sobolev Coefficientsmentioning

confidence: 99%

Some results on Dunkl-coherent Pairs

Sghaier

Hamdi

2020

Integral Transforms and Special Functions

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“…In the framework of coherent pairs of measures, it is possible to obtain the associated Sobolev–Fourier coefficients with low computational cost (see Ref. [14]). The convergence of the corresponding Sobolev–Fourier expansions for the Jacobi weights is analyzed in Refs.…”

Section: Introductionmentioning

confidence: 99%

Coherent pairs of measures of the second kind on the real line and Sobolev orthogonal polynomials. An application to a Jacobi case

et al. 2023

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The aim here is to consider the orthogonal polynomials  𝑛 (𝜈 0 , 𝜈 1 , 𝑠; 𝑥) with respect to an inner product of the type ⟨𝑓, 𝑔⟩ 𝔰 = ∫ 𝑓(𝑥)𝑔(𝑥)𝑑𝜈 0 (𝑥) + 𝑠 ∫ 𝑓 ′ (𝑥)𝑔 ′ (𝑥)𝑑𝜈 1 (𝑥), where 𝑠 > 0 and {𝜈 0 , 𝜈 1 } is a coherent pair of positive measures of the second kind on the real line (CPPM2K on the real line). Properties of  𝑛 (𝜈 0 , 𝜈 1 , 𝑠; 𝑥) and the connection formulas they satisfy with the orthogonal polynomials associated with the measure 𝜈 0 are analyzed. It is also shown that the zeros of  𝑛 (𝜈 0 , 𝜈 1 , 𝑠; 𝑥) are the eigenvalues of a matrix, which is a single line modification of the 𝑛 × 𝑛 Jacobi matrix associated with the measure 𝜈 0 . The paper also looks at a special example of a CPPM2K on the real line, where one of the measures is the Jacobi measure, and provides a much more detailed study of the properties of the orthogonal polynomials and the corresponding connection coefficients. In particular, the relation that these connection coefficients have with the Wilson polynomials is exposed.

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Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics

Molano

2024

Ukr Math J

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On Symmetric (1, 1)-Coherent Pairs and Sobolev Orthogonal polynomials: an algorithm to compute Fourier coefficients

Cited by 5 publications

References 12 publications

Some results on Dunkl-coherent Pairs

Some results on Dunkl-coherent Pairs

Coherent pairs of measures of the second kind on the real line and Sobolev orthogonal polynomials. An application to a Jacobi case

Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics

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