Characterization and construction of classical orthogonal polynomials using a matrix approach

Verde-Star, Luis

doi:10.1016/j.laa.2013.01.014

Cited by 21 publications

(21 citation statements)

References 8 publications

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“….Þ and B I. The statement of Lemma 3.1 generalizes (4.2) in [22] for families of orthogonal polynomials related by a relation like p 0 ðxÞ MqðxÞ.…”

Section: A Matrix Interpretation Of ðM; Nþ-coherencementioning

confidence: 58%

“…On the other hand, the well known Hahn's condition characterizing classical orthogonal polynomials (see [6]) has been considered from a matrix approach in [22] where nice and elegant proofs of some well known characterizations of classical orthogonal polynomials in terms of the Jacobi matrices are given. Therein, the connection between the Jacobi matrices associated with the SMOP fP n ðxÞg nP0 and f P 0 n ðxÞ nþ1 g nP0 is stated in formula (4.2) and, as a consequence, it is possible to get the general expression of the coefficients of the TTRR for classical orthogonal polynomials by identifying the cor responding entries.…”

Section: A Matrix Interpretation Of ðM; Nþ-coherencementioning

confidence: 99%

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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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“….Þ and B I. The statement of Lemma 3.1 generalizes (4.2) in [22] for families of orthogonal polynomials related by a relation like p 0 ðxÞ MqðxÞ.…”

Section: A Matrix Interpretation Of ðM; Nþ-coherencementioning

confidence: 58%

Section: A Matrix Interpretation Of ðM; Nþ-coherencementioning

confidence: 99%

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

Marcellán

Pinzón-Cortés

2014

Applied Mathematics and Computation

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“…The proof can be found in [20]. Notice that this is a matrix version of the Favard's theorem, and the entries of J, i.e., the coefficients of the recurrence relation for the orthogonal polynomials, can be obtained from the matrix A.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, in [20], a matrix characterization for classical orthogonal polynomials was introduced. Let write P n ðxÞ X n j 0 a n;j x j ; n P 0;…”

Section: Introductionmentioning

confidence: 99%

“…Following the notation used in [20], we say that a matrix B is a lower semi matrix if there exists an integer m such that b i;j 0 whenever i j < m. The entry b i;j is in the mth diagonal i j m. If B is non zero, we say that B has index m, indðBÞ m, if m is the minimum integer such that B has at least one nonzero entry in the mth diagonal, also if all the entries in its diagonal of index m are equal to 1; B is called monic. Finally, B is said to be ðn; mÞ banded if there exists a pair of inte gers ðn; mÞ with n 6 m and all the nonzero entries of B lie between the diagonals of indices n and m. It is easy to see that the set of banded matrices is closed under addition and multiplication, despite the fact that the inverse of a banded matrix might not be banded.…”

Section: Introductionmentioning

confidence: 99%

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