Sobolev orthogonal polynomials and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si28.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-coherent pairs of measures

Jesus, M. N. de; Petronilho, J.

doi:10.1016/j.cam.2012.07.006

Cited by 17 publications

(2 citation statements)

References 19 publications

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“…The notion of "(M, N)-coherence" was introduced in [14] for order one, where the natural connection with Sobolev polynomials orthogonal with respect to the inner product (1) is presented.…”

Section: Introductionmentioning

confidence: 99%

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

2019

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“…The notion of "(M, N)-coherence" was introduced in [14] for order one, where the natural connection with Sobolev polynomials orthogonal with respect to the inner product (1) is presented.…”

Section: Introductionmentioning

confidence: 99%

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

2019

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“…Later, it has been very fruitful in the study of algebraic and asymptotic properties of Sobolev orthogonal polynomials. In fact, the original concept was generalized on several occasions (for example, in [8] or [9]), being the most general the one introduced in 2014 with the (M, N)-coherent pairs of order (m, k) (see [10]).…”

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confidence: 99%

Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

2020

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In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential operator, we are interested in the corresponding eigenvalues, more exactly, in their asymptotic behavior. Thus, we can determine a limit value which links this asymptotic behavior and the uniform norm of the orthonormal polynomials in a logarithmic scale. This value appears in the theory of reproducing kernel Hilbert spaces. On the other hand, we tackle a more general case than the one considered in the literature previously.

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(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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Sobolev orthogonal polynomials and (M,N)-coherent pairs of measures

Cited by 17 publications

References 19 publications

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

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

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