×
Maroni, Pascal; Mesquita, Teresa A.

Cubic decomposition of 2-orthogonal polynomial sequences. (English) Zbl 1266.42069

Mediterr. J. Math. 10, No. 2, 843-863 (2013).
From the text: We study the general cubic decomposition (CD) of a two-orthogonal polynomial sequence, beginning with a characterization of all the elements involved in such CD. The recurrence coefficients of the two-orthogonal sequences which admit a diagonal CD are described and we prove that the correspondent principal components are also two-orthogonal. Finally, we analyze the CD of a two-symmetric and two-orthogonal sequence.

Cited in 4 Documents

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

two-orthogonal polynomials; finite-type relations; cubic decomposition
Cite Review PDF
Full Text: DOI

References:

[1] Barrucand P., Dickinson D.: On cubic transformation of orthogonal polynomials. Proc. Amer. Math. Soc. 17, 810–814 (1966) · Zbl 0143.08601 · doi:10.1090/S0002-9939-1966-0197805-1
[2] Y. Ben Cheikh, On some (n 1)-symmetric linear functionals J. Comput. Appl. Math. 133 (2001), 207–218. · Zbl 1008.42020
[3] D. Bessis and P. Moussa, Orthogonality properties of iterated polynomial mappings, Comm. Math. Phys. 88 (1983), no. 4, 503–529. · Zbl 0523.30019
[4] J. A. Charris and M. E. H. Ismail, Sieved orthogonal polynomials. VII: Generalized polynomial mappings, Trans. Am. Math. Soc., Vol. 340, No. 1 (Nov., 1993), 71–93. · Zbl 0790.33007
[5] J. A. Charris, M. E. H. Ismail and S. Monsalve, On sieved orthogonal polynomials. X. General blocks of recurrence relations, Pacific J. Math. 163 (1994), no. 2, 237–267. · Zbl 0805.33006
[6] Chihara T.S.: On kernel polynomials and related systems. Boll. Un. Mat. Ital. 19(3), 451–459 (1964) · Zbl 0125.31406
[7] K. Douak and P.Maroni, Les polynômes orthogonaux ”classiques” de dimension deux, Analysis 12 (1992), 71–107. · Zbl 0767.33004
[8] K. Douak and P. Maroni, On d-orthogonal Tchebyshev polynomials, I, Appl. Numer. Math. 24 (1997), 23–53. · Zbl 0881.33024
[9] J. S. Geronimo andW. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Am. Math. Soc., Vol. 308, No. 2 (Aug., 1988), 559–581. · Zbl 0652.42009
[10] I. Rodriguez González and C. Tasis Montes, Descomposition cubica general de una sucesion de polinomios ortogonales, Actas del Simposium Polinomios Ortogonales y Aplicaciones, Gijón, 1989, 259–265.
[11] M. E. H. Ismail, On sieved orthogonal polynomials. III: Orthogonality on several intervals, Trans. Am. Math. Soc., Vol. 294, No. 1 (Mar., 1986), 89–111. · Zbl 0612.33009
[12] de Jesus M.N., Petronilho J.: On orthogonal polynomials obtained via polynomial mappings. J. Approx. Theory 162, 2243–2277 (2010) · Zbl 1209.42018 · doi:10.1016/j.jat.2010.07.012
[13] A. Macedo and P. Maroni, General quadratic decomposition, J. Difference Equ. Appl. 16, no. 11 (2010), 1309 –1329. · Zbl 1203.42037
[14] Marcellán F., Sansigre G.: Orthogonal polynomials and cubic transformations. J. Comput. Appl. Math. 49, 161–168 (1993) · Zbl 0790.42010 · doi:10.1016/0377-0427(93)90147-4
[15] F. Marcellán and J. Petronilho, Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl. 260 (1997), 169–208. · Zbl 0877.15010
[16] F. Marcellán and J. Petronilho, Orthogonal polynomials and quadratic transformations, Port. Math. (N.S.) 56 (1999), 81–113. · Zbl 0936.42012
[17] Marcellán F., Petronilho J.: Orthogonal polynomials and cubic polynomial mappings. I. Commun. Anal. Theory Contin. Fract. 8, 88–116 (2000)
[18] Marcellán F., Petronilho J.: Orthogonal polynomials and cubic polynomial mappings. II. The positive–definite case. Commun. Anal. Theory Contin. Fract. 9, 11–20 (2001)
[19] P. Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieur á deux, Ann. Fac. Sci. Toulouse 10 (1), (1989), 105–139.
[20] P. Maroni, Sur la décomposition quadratique d’une suite de polynômes orthogonaux I, Rivista di Mat. Pura ed Appl. 6 (1990), pp. 19–53. · Zbl 0715.33009
[21] P. Maroni, Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math. 48 (1993), 133–155. · Zbl 0790.33006
[22] P. Maroni, Semi-classical character and finite-type relations between polynomial sequences, Appl. Numer. Math. 31 (1999), 295–330. · Zbl 0962.42017
[23] Maroni P., Mesquita T.A., da Rocha Z.: On the general cubic decomposition of polynomial sequences. J. Difference Equ. Appl. 17(9), 1303–1332 (2011) · Zbl 1232.33017 · doi:10.1080/10236190903527236
[24] T. A. Mesquita, Polynomial Cubic Decomposition. Ph.D. Thesis, Universidade do Porto, Faculdade de Ciências, Departamento de Matemática, 2010.
[25] J. Van Iseghem, Approximants de Padé vectoriels, Thèse d’état, Univ. des Sciences et techiques de Lille–Flandre–Artois, 1987.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.