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

Petronilho, J.

doi:10.1016/j.jmaa.2005.05.018

Cited by 31 publications

(26 citation statements)

References 12 publications

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“…where a i,n and b i,n are complex numbers such that The next theorem improves several results stated in [14,15,17,13] by giving a complete description of the semiclassical case in the framework of (M, N)-coherence of order (m, k).…”

Section: Theorem 25 ([242519]) a Regular Linear Functionalsupporting

confidence: 62%

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(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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Section: Theorem 25 ([242519]) a Regular Linear Functionalsupporting

confidence: 62%

“…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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“…, β N−1 are constants. Taking the kth derivative on both sides of (20) and taking into account the relations…”

Section: Resultsmentioning

confidence: 99%

On linearly related sequences of derivatives of orthogonal polynomials

Jesus

Petronilho

2008

Journal of Mathematical Analysis and Applications

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We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (P n ) n and (Q n ) n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such asfor all n = 0, 1, 2, . . . , where M and N are fixed nonnegative integer numbers, and r i,n and s i,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (P n ) n and (Q n ) n (resp.). Assuming 0 m k, we prove the existence of four polynomials Φ M+m+i and Ψ N+k+i , of degrees M + m + i and N + k + i (resp.), such thatthe (k − m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k = m, then u and v are connected by a rational modification. If k = m + 1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k > m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k − m with polynomial coefficients.

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“…The above definition unifies the generalizations given in the literature of the concept of coherent pair (in our terminology, ð1; 0Þ coherent pair) introduced by Iserles, Koch, Nørsett, and Sanz Serna in [8] (see also the work of Meijer [19]). In fact, Delgado and Marcellán [5] introduced ð1; 1Þ coherent pairs, Kwon, Lee, and Marcellán [12] considered ð2; 0Þ coherent pairs, Maroni and Sfaxi [17,18] looked at the ðM þ N; 2M þ 1Þ coherence relation, Marcellán, Martínez Finkelshtein, and Moreno Balcázar [14] presented the ðM; 0Þ coherence relation, Alfaro, Marcellán, Peña, and Rezola [1,2] studied ð1; 1Þ coherent pairs of order 0, Petronilho [20] considered the ðM; NÞ coherence relation of order 0, de Jesus and Petronilho [10,11] analyzed ðM; NÞ coherent pairs, Branquinho and Rebocho [3] looked at the ð1; 0Þ coherence relation of order 2, and, Marcellán and Pinzón Cortés [15] studied ð1; 1Þ coherent pairs of order m. For a review about these and other related works, see for instance, the introductory sections in the papers [15,11].…”

Section: Introductionmentioning

confidence: 99%

(M,N)-Coherent pairs of linear functionals and Jacobi matrices

Marcellán

Pinzón-Cortés

2014

Applied Mathematics and Computation

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Keywords:Coherent pairs Structure relations Regular linear functionals Orthogonal polynomials Classical orthogonal polynomials Sobolev orthogonal polynomials Monic Jacobi matrix a b s t r a c t A pair of regular linear functionals ðU; VÞ in the linear space of polynomials with complex coefficients is said to be an ðM; NÞ coherent pair of order m if their corresponding sequences of monic orthogonal polynomials fP n ðxÞg nP0 and fQ n ðxÞg nP0 satisfy a structure relationwhere M; N, and m are non negative integers, fa i;n g nP0 ; 0 6 i 6 M, and fb i;n g nP0 ; 0 6 i 6 N, are sequences of complex numbers such that a M;n -0 if n P M; b N;n -0 if n P N, andVÞ is called an ðM; NÞ coherent pair.In this work, we give a matrix interpretation of ðM; NÞ coherent pairs of linear function als. Indeed, an algebraic relation between the corresponding monic tridiagonal (Jacobi) matrices associated with such linear functionals is stated. As a particular situation, we ana lyze the case when one of the linear functionals is classical. Finally, the relation between the Jacobi matrices associated with ðM; NÞ coherent pairs of linear functionals of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the Sobolev inner product defined by the pair ðU; VÞ is deduced.

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

Cited by 31 publications

References 12 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 sequences of derivatives of orthogonal polynomials

(M,N)-Coherent pairs of linear functionals and Jacobi matrices

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