When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Alfaro, M.; Marcellán, Francisco; Peña, Ana; Rezola, M. L.

doi:10.1016/j.cam.2009.02.060

Cited by 31 publications

(27 citation statements)

References 9 publications

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“…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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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: 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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“…The above relations provide a complete characterization of the orthogonality of the polynomial sequence {Q n } n 0 . When b j,n = b j , j = 1, · · · , k − 1, you recover Theorem 1 in [3].…”

Section: )mentioning

confidence: 74%

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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“…The comparison with the results given in [4,2] shows that the periodic character of the coefficients of the three-term recurrence relation pointed out in (i) and (ii) (a) can be expected but the coefficients of the three-term recurrence relation obtained in (ii) (b) are apparently unexpected. An interesting problem is the analysis of the integral representation of the linear functionals associated with the SMOP such that the coefficients of the three-term recurrence relation are described in (ii) (b).…”

mentioning

confidence: 70%

“…These orthogonality conditions are also obtained in [4]. Now, our aim is to define all SMOP fP n } n$0 such that fQ n } n$0 given by (3.1) is also orthogonal.…”

Section: Special Casesmentioning

confidence: 99%

“…For the case m 2, k 2, one can find a detailed investigation in [3]. Later on, the situation m 1, k N, with constant coefficients has been considered in [4]. In this contribution, necessary and sufficient conditions in order for fQ n } n$0 to be an SMOP are obtained, and a matrix method by using Jacobi matrices associated with fP n } n$0 and fQ n } n$0 is derived.…”

Section: Introductionmentioning

confidence: 99%

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On an inverse problem for a linear combination of orthogonal polynomials

Marcellán

Varma

2013

Journal of Difference Equations and Applications

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This paper deals with the analysis of the orthogonality of a monic polynomial sequence fQ n } n$0 defined as a linear combination of a sequence of monic orthogonal polynomials fP n } n$0 withwhere r n -0 for n $ 3. Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials fQ n } n$0 . We obtain the connection between the Jacobi matrices associated with fP n } n$0 and fQ n } n$0 , respectively, by using an LU factorization. Some special cases of the above type relation are analysed.

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When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Cited by 31 publications

References 9 publications

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

(M,N)-coherent pairs of order(m,k)and Sobolev orthogonal

Orthogonality of quasi-orthogonal polynomials

On an inverse problem for a linear combination of orthogonal polynomials

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