On Kernel polynomials and self-perturbation of orthogonal polynomials

Kwon, Kil Hyun; Lee, D.W.; Marcellán, Francisco; Park, S.B.

doi:10.1007/s10231-001-8200-7

Cited by 14 publications

(5 citation statements)

References 13 publications

(7 reference statements)

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“…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]. Theorem 1.1 gives a solution for the inverse problem associated to (1).…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

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: 97%

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

Petronilho

2006

Journal of Mathematical Analysis and Applications

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“…Remark 2.5. Notice that s n = 0 for all n¿2 and t 3 = 0 yields, according to Theorem 4:7 in [6], n = ÿ n ; n¿0; n = n−2 ; n = n ; n¿2;…”

Section: Proof ⇐) Trivialmentioning

confidence: 95%

“…The situation R n (x) ≡ Q n (x) is considered in [6] where necessary and su cient conditions for the orthogonality of the sequence {P n (x)} ∞ n=0 are obtained. If t = 1 and R n (x) = Q (1) n (x), i.e., the sequence of monic associated polynomials of the ÿrst kind, then for n = =constant we obtain the so-called sequence of co-recursive monic orthogonal polynomials (see [3,4,10,11]).…”

Section: Introductionmentioning

confidence: 99%

Orthogonality of linear combinations of two orthogonal polynomial sequences

Kwon¹,

Lee²,

Marcellán³

2001

Journal of Computational and Applied Mathematics

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“…Observe that we adopt the convention that when either m or k is equal to one, then the corresponding sum does not appear, that is, we interpret it as an empty one. A vast number of interesting results have been obtained on topics related to the inverse problem (see [1,3,4,5,10,11,25,26,32,37]).…”

Section: Orthogonality Of Quasi-orthogonal Polynomialsmentioning

confidence: 99%

Orthogonality of quasi-orthogonal polynomials

Bracciali¹,

Marcellán²,

Varma³

2018

Filomat

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A result of Pólya states that every sequence of quadrature formulas Qn(f ) with n nodes and positive numbers converges to the integral I(f ) of a continuous function f provided Qn(f ) = I(f ) for a space of algebraic polynomials of certain degree that depends on n. The classical case when the algebraic degree of precision is the highest possible is well-known and the quadrature formulas are the Gaussian ones whose nodes coincide with the zeros of the corresponding orthogonal polynomials and the numbers are expressed in terms of the so-called kernel polynomials. In many cases it is reasonable to relax the requirement for the highest possible degree of precision in order to gain the possibility to either approximate integrals of more specific continuous functions that contain a polynomial factor or to include additional fixed nodes. The construction of such quadrature processes is related to quasi-orthogonal polynomials. Given a sequence {Pn} n 0 of monic orthogonal polynomials and a fixed integer k, we establish necessary and sufficient conditions so that the quasi-orthogonal polynomials {Qn} n 0 defined bywith b i,n ∈ R, and b k−1,n = 0 for n k − 1, also constitute a sequence of orthogonal polynomials. Therefore we solve the inverse problem for linearly related orthogonal polynomials. The characterization turns out to be equivalent to some nice recurrence formulas for the coefficients b i,n . We employ these results to establish explicit relations between various types of quadrature rules from the above relations. A number of illustrative examples are provided.2010 Mathematics Subject Classification. 33C45; 42C05. Key words and phrases. Orthogonal polynomials, quasi-orthogonal polynomials, positive quadrature formulas, Gaussian quadrature formulas, Christoffel numbers, inverse problems.

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On Kernel polynomials and self-perturbation of orthogonal polynomials

Cited by 14 publications

References 13 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

Orthogonality of linear combinations of two orthogonal polynomial sequences

Orthogonality of quasi-orthogonal polynomials

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