On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

Sánchez-Lara, J.

doi:10.1216/rmj-2017-47-1-267

Cited by 6 publications

(7 citation statements)

References 18 publications

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“…Observe that B acts on polynomials of degree ≤ 2m −1 only through D, and, for polynomials of degree ≥ 2m, B acts only through C. In the same way, in [19], the Jacobi polynomials…”

Section: Introductionmentioning

confidence: 93%

“…Special attention has been given to the case when u N is associated with the classical Jacobi or Laguerre orthogonal polynomials, and u k , 0 ≤ k ≤ N − 1, are derivatives of Dirac deltas supported on the endpoints of the interval of orthogonality of u N (see [1][2][3]5,15,18,19] and the references therein). Such bilinear forms arise when studying the orthogonality of Jacobi or Laguerre polynomials with the so-called non standard parameters, that is, negative integer parameters such that the coefficient c n in the corresponding three-term recurrence relation x p n (x) = a n p n+1 (x) + b n p n (x) + c n p n−1 (x), vanishes.…”

Section: Introductionmentioning

confidence: 99%

“…Using some ideas from [17], a construction of discrete-continuous bilinear forms is presented in [19], that is valid for general orthogonal polynomials { p n (x)} n≥0 , not necessarily the Jacobi and Laguerre polynomials. Therein, the discrete part D is defined by D( p n , p m ) = k n δ n,m , n, m ≤ N − 1, with arbitrary positive constants k n .…”

Section: Introductionmentioning

confidence: 99%

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On Sobolev bilinear forms and polynomial solutions of second-order differential equations

García-Ardila

Marriaga

2021

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Given a linear second-order differential operator $${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$ L ≡ ϕ D 2 + ψ D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$\lambda _n$$ λ n , $$n\geqslant 0$$ n ⩾ 0 , and a Sobolev bilinear form $$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$ B ( p , q ) = ∑ k = 0 N u k , p ( k ) q ( k ) , N ⩾ 0 , where $${\mathbf {u}}_k$$ u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${\mathcal {L}}[y]=\lambda _n\,y$$ L [ y ] = λ n y with respect to $${\mathcal {B}}$$ B . We show that such polynomials are orthogonal with respect to $${\mathcal {B}}$$ B if the Pearson equations $$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$ D ( ϕ u k ) = ( ψ + k ϕ ′ ) u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

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“…Observe that B acts on polynomials of degree ≤ 2m −1 only through D, and, for polynomials of degree ≥ 2m, B acts only through C. In the same way, in [19], the Jacobi polynomials…”

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

García-Ardila

Marriaga

2021

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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“…Notice that, for a symmetric inner product of type (1), we cannot derive the asymptotic behavior of the corresponding eigenvalues n from Theorem 1 in a straightforward way. This is because the expressions for n are essentially different in each case (see (16) and (17)). Thus, as we have commented previously, the symmetric case is technically easier to analyze and it was done in [31].…”

Section: Remarkmentioning

confidence: 99%

“…On the other hand, the discrete-continuous case attracted interest when it was discovered that this type of inner product provides Sobolev orthogonality to families of classical orthogonal polynomials with nonstandard parameters (Gegenabauer, Jacobi, and Laguerre polynomials), see details in the papers [11][12][13][14][15] or in a more general framework [16].…”

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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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On the Relation Between Gegenbauer Polynomials and the Ferrers Function of the First Kind

Cohl

Costas-Santos

2022

Anal Math

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On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

Cited by 6 publications

References 18 publications

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

On the Relation Between Gegenbauer Polynomials and the Ferrers Function of the First Kind

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