On rational transformations of linear functionals: direct problem

Alfaro, M.; Marcellán, Francisco; Peña, Ana; Rezola, M. L.

doi:10.1016/j.jmaa.2004.04.065

Cited by 18 publications

(35 citation statements)

References 10 publications

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“…Rezola [1], where the authors characterized linearly related sequences of orthogonal polynomials involving linear combinations with exactly two elements-corresponding to the case M = N = 1 in (1)-in terms of their functionals. Further, in [2] the above authors gave a complete discussion of the regularity conditions involving a rational modification as (x − a)u = λ(x − b)v between two moment linear functionals u and v (thus, corresponding again to the case M = N = 1). Special cases of these type of relations were treated in [7,9,14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On the linear functionals associated to linearly related sequences of orthogonal polynomials

Petronilho

2006

Journal of Mathematical Analysis and Applications

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An inverse problem is solved, by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials (P n ) n and (Q n ) n , respectively, in the sensefor all n = 0, 1, 2, . . . (where r i,n and s i,n are complex numbers satisfying some natural conditions), are connected by a rational modification, i.e., there exist polynomials φ and ψ, with degrees M and N , respectively, such that φu = ψv. We also make some remarks concerning the corresponding direct problem, stating a characterization theorem in the case N = 1 and M = 2. As an example, we give a linear relation of the above type involving Jacobi polynomials with distinct parameters.  2005 Elsevier Inc. All rights reserved.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This case was completely solved in [1,2]. Following the proof of Theorem 5.1, we can state the next proposition, which is a slightly modification of Theorem 2.4 in [1] (see also [16]).…”

Section: The Case M = N =mentioning

confidence: 95%

On the linear functionals associated to linearly related sequences of orthogonal polynomials

Petronilho

2006

Journal of Mathematical Analysis and Applications

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“…Meijer [6], M. Alfaro, F. Marcellán, A. Peña, and M.L. Rezola [7][8][9][10], A.M. Delgado and F. Marcellán [11,12], J. Petronilho [13], M.N. de Jesus and J. Petronilho [14,15], A. Branquinho and M.N.…”

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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“…Moreover, P. Maroni [13] gave a characterization of the relation between the MOPSs associated with two regular functionals u and v fulfilling Φu = Ψv. In connection with the study of direct problems related to orthogonal polynomials associated with this kind of modifications of linear functionals (rational modifications), besides the work [2], among others we also point out the works by W. Gautschi [8], M. Sghaier and J. Alaya [15], M.I. Bueno and F. Marcellán [5], and J.H.…”

Section: Introductionmentioning

confidence: 86%

Orthogonal polynomials generated by a linear structure relation: Inverse problem

Alfaro

Peña

Petronilho

et al. 2013

Journal of Mathematical Analysis and Applications

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Let (Pn)n and (Qn)n be two sequences of monic polynomials linked by a type structure relation such aswhere (rn)n, (sn)n and (tn)n are sequences of complex numbers.Firstly, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences (Pn)n and (Qn)n are orthogonal with respect to regular moment linear functionals u and v, respectively.Secondly, assuming that the above relation is non-degenerate and (Pn)n is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence (Qn)n in terms of the coefficients of the polynomials Φ and Ψ which appear in the rational transformation (in the distributional sense) Φu = Ψv .Some illustrative examples of the developed theory are presented.2000 Mathematics Subject Classification. 42C05, 33C45.

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On rational transformations of linear functionals: direct problem

Abstract: Let u be a quasi-definite linear functional. We find necessary and sufficient conditions in order to the linear functional v satisfying (x −ã)u = λ(x − a)v be a quasi-definite one. Also we analyze some linear relations linking the polynomials orthogonal with respect to u and v.

Cited by 18 publications

References 10 publications

On the linear functionals associated to linearly related sequences of orthogonal polynomials

On the linear functionals associated to linearly related sequences of orthogonal polynomials

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

Orthogonal polynomials generated by a linear structure relation: Inverse problem

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