Abstract
Let \(\{\mathbb{P}_{n}\}_{n\ge 0}\) and \(\{\mathbb{Q}_{n}\}_{n\ge 0}\) be two monic polynomial systems in several variables satisfying the linear structure relation \(\mathbb{Q}_{n} = \mathbb{P}_{n} + M_{n} \mathbb{P}_{n-1}, \quad n\ge 1,\)where M n are constant matrices of proper size and \(\mathbb{Q}_{0} = \mathbb{P}_{0}\). The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.
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References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. 9th printing. Dover, New York (1972)
Alfaro, M., Marcellán, F., Peña, A., Rezola, M.L.: On linearly related orthogonal polynomials and their functionals. J. Math. Anal. Appl. 287, 307–319 (2003)
Alfaro, M., Marcellán, F., Peña, A., Rezola, M.L.: When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials. J. Comput. Appl. Math. 233, 1446–1452 (2010)
Alfaro, M., Peña, A., Petronilho, J., Rezola, M.L.: Orthogonal polynomials generated by a linear structure relation: inverse problem. J. Math. Anal. Appl. 401, 182–197 (2013)
Alfaro, M., Peña, A., Rezola, M.L., Marcellán, F.: Orthogonal polynomials associated with an inverse quadratic spectral transform. Comput. Math. Appl. 61, 888–900 (2011)
Beardon, H., Driver, K.A.: The zeros of linear combinations of orthogonal polynomials. J. Approx. Theory 137, 179–186 (2005)
Berens, H., Schmid, H., Xu, Y.: Multivariate Gaussian cubature formula. Arch. Math. 64, 26–32 (1995)
Brezinski, C., Driver, K.A., Redivo–Zaglia, M.: Quasi–orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157–168 (2004)
Chihara, T.S.: An introduction to orthogonal polynomials. Gordon and Breach, New York (1978)
Delgado, A.M., Fernández, L., Pérez, T.E., Piñar, M.A.: On the Uvarov modification of two variable orthogonal polynomials on the disk. Complex Anal. Oper. Theory 6, 665–676 (2012)
Delgado, A.M., Fernández, L., Pérez, T.E., Piñar, M.A., Xu, Y.: Orthogonal polynomials in several variables for measures with mass points. Numer. Algorithms 55, 245–264 (2010)
Delgado, A.M., Geronimo, J.S., Iliev, P., Xu, Y.: On a two–variable class of Bernstein–Szegő Measures. Constr. Approx. 30, 71–91 (2009)
Driver, K.A., Jordaan, K.: Zeros of linear combinations of Laguerre polynomials from different sequences. J. Comput. Appl. Math. 233, 719–722 (2009)
Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. Cambridge University Press, Cambridge (2001)
Fernández, L., Pérez, T.E., Piñar, M.A., Xu, Y.: Krall–type orthogonal polynomials in several variables. J. Comput. Appl. Math. 233, 1519–1524 (2010)
Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1985)
Iserles, A., Koch, P.E., Nørsett, S.P., Sanz–Serna, J.M.: On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65, 151–175 (1991)
Koornwinder, T.H.: Two variable analogues of the classical orthogonal polynomials. In: Askey, R. (ed.) Theory and application of special functions, pp. 435–495. Academic Press, New York (1975)
Marcellán, F., Maroni, P.: Sur l’adjonction d’une masse de Dirac à une forme régulière et semi–classique. Ann. Mat. Pura Appl. 162, 1–22 (1992)
Marcellán, F., Petronilho, J.: Orthogonal polynomials and coherent pairs: the classical case. Indag. Math. (N.S.) 6, 287–307 (1995)
Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Applications aux polynômes orthogonaux semi–classiques. IMACS Ann. Comput. Appl. Math. 9, 95–130 (1991)
Meijer, H.G.: Determination of all coherent pairs. J. Approx. Theory 89, 321–343 (1997)
Morrow, C.R., Patterson, T.N.L.: Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal. 15, 953–976 (1978)
Mysovskikh, I.P. Interpolation cubature formulas. (In Russian) Nauka, Moscow (1981)
Pérez, T.E., Piñar, M.A., Xu, Y.: Weighted Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory 171, 84–104 (2013)
Schmid, H.: On cubature formulae with a minimum number of knots. Numer. Math. 31, 281–297 (1978)
Schmid, H., Xu, Y.: On bivariate Gaussian cubature formulae. Proc. Amer. Math. Soc. 122, 833–841 (1994)
Shohat, J.A.: On mechanical quadratures, in particular, with positive coefficients. Trans. Am. Math. Soc. 42, 461–496 (1937)
Szegő, G.: Orthogonal Polynomials, vol. 23, 4th edn. American Mathematical Social Colloquium Publishing, Providence (1978)
Xu, Y.: Gaussian cubature and bivariate polynomial interpolation. Math. Comp. 59, 547–555 (1992)
Xu, Y.: On zeros of multivariate quasi–orthogonal polynomials and Gaussian cubature formulae. SIAM J. Math. Anal. 25, 991–1001 (1994)
Xu, Y.: A family of Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory 138, 232–241 (2006)
Zhedanov, A.: Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85, 67–86 (1997)
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Manuel Alfaro, Ana Peña, M. Luisa Rezola were partially supported by MEC of Spain under Grant MTM2012–36732–C03–02 and Diputación General de Aragón project E–64.
Teresa E. Pérez was partially supported by MICINN of Spain and by the European Regional Development Fund (ERDF) through the grant MTM2011–28952–C02–02, and Junta de Andalucía FQM–0229, P09–FQM–4643 and P11-FQM-7276.
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Alfaro, M., Peña, A., Pérez, T.E. et al. On linearly related orthogonal polynomials in several variables. Numer Algor 66, 525–553 (2014). https://doi.org/10.1007/s11075-013-9747-2
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DOI: https://doi.org/10.1007/s11075-013-9747-2