Orthogonal polynomials generated by a linear structure relation: Inverse problem

Alfaro, M.; Peña, Ana; Petronilho, J.; Rezola, M. L.

doi:10.1016/j.jmaa.2012.12.004

Cited by 10 publications

(12 citation statements)

References 16 publications

(44 reference statements)

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“…Rebocho [16], F. Marcellán and N.C. Pinzón-Cortés [17], and M. Alfaro, A. Peña, J. Petronilho, and M.L. Rezola [18]. For a review about these and other contributions, see e.g.…”

Section: Introductionmentioning

confidence: 99%

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

Jesus¹,

Marcellán²,

Petronilho³

et al. 2014

Journal of Computational and Applied Mathematics

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a b s t r a c tA pair of regular linear functionals (U, V) is said to be a (M, N)-coherent pair of order (m, k) if their corresponding sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 satisfy a structure relation such as To be more precise, let us assume that m ≥ k. If m = k then U and V are related by a rational factor (in the distributional sense); if m > k then U and V are semiclassical and they are again related by a rational factor. In the second part of this work we deal with a Sobolev type inner product defined in the linear space of polynomials with real coefficients, P, aswhere λ is a positive real number, m is a positive integer number and (μ 0 , μ 1 ) is a (M, N)-coherent pair of order m of positive Borel measures supported on an infinite subset of the real line, meaning that the sequences of monic orthogonal polynomials {P n (x)} n≥0 and {Q n (x)} n≥0 with respect to μ 0 and μ 1 , respectively, satisfy a structure relation as above with k = 0, a i,n and b i,n being real numbers fulfilling the above mentioned conditions. We generalize several recent results known in the literature in the framework of Sobolev orthogonal polynomials and their connections with coherent pairs (introduced in [A. Iserles, P.E. Koch, S.P. Nørsett, J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products J. Approx. Theory 65 (2) (1991) 151-175]) and their extensions. In particular, we show how to compute the coefficients of the Fourier expansion of functions on an appropriate Sobolev space (defined by the above inner product) in terms of the sequence of Sobolev orthogonal polynomials {S n (x; λ)} n≥0 .

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“…Rebocho [16], F. Marcellán and N.C. Pinzón-Cortés [17], and M. Alfaro, A. Peña, J. Petronilho, and M.L. Rezola [18]. For a review about these and other contributions, see e.g.…”

Section: Introductionmentioning

confidence: 99%

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

Jesus¹,

Marcellán²,

Petronilho³

et al. 2014

Journal of Computational and Applied Mathematics

Self Cite

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“…As it was proven in [4], the moment functional v x is quasi-definite if and only if either α n := Γ(α + 1)Γ(α + β + 2)Γ(n)Γ(n + β) + M Γ(β + 1)Γ(n + α)Γ(n + α + β) = 0, n ≥ 2, for α = 0, and M := − 2(β + 1) + a 1 (α + β + 1)(α + β + 2) 2(α + 1) + a 1 (α + β + 2) , or α n := 2(β + 2) 2 + a 1 (β + 2) − (β + 1)…”

Section: Krall Jacobi-jacobi Orthogonal Polynomialsmentioning

confidence: 63%

“…Lemma 1 Let {P n } n≥0 and {Q n } n≥0 be two monic orthogonal polynomial systems linearly related by (4). Then (i) If rank M 1 = 0, then rank M n = 0 for every n ≥ 1, (ii) If rank M 1 = 1, then rank M n = r d n−1 for every n ≥ 1.…”

Section: Resultsmentioning

confidence: 99%

On linearly related orthogonal polynomials in several variables

et al. 2013

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Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$ are constant matrices of proper size and $\mathbb{Q}_0 = \mathbb{P}_0$. The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.Comment: 28 pages. To appear in Numerical Algorithm

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“…First time, this type of inverse problem was discussed in [5]. Recently, similar analysis has been done in [1] for the case m 2, k 3. On the other hand, the general linear structure relations given by (1.2) have been analysed in [22] with an additional assumption about the orthogonality of fQ n } n$0 .…”

Section: Introductionmentioning

confidence: 94%

On an inverse problem for a linear combination of orthogonal polynomials

Marcellán

Varma

2013

Journal of Difference Equations and Applications

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This paper deals with the analysis of the orthogonality of a monic polynomial sequence fQ n } n$0 defined as a linear combination of a sequence of monic orthogonal polynomials fP n } n$0 withwhere r n -0 for n $ 3. Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials fQ n } n$0 . We obtain the connection between the Jacobi matrices associated with fP n } n$0 and fQ n } n$0 , respectively, by using an LU factorization. Some special cases of the above type relation are analysed.

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Orthogonal polynomials generated by a linear structure relation: Inverse problem

Cited by 10 publications

References 16 publications

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

On linearly related orthogonal polynomials in several variables

On an inverse problem for a linear combination of orthogonal polynomials

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