×

On Sobolev orthogonal polynomials. (English) Zbl 1351.33011

This very valuable review article summarizes the results and development in the field of the Sobolev orthogonal polynomials. The Sobolev orthogonal polynomials of one variable are polynomials orthogonal with respect to the scalar product of the general form \[ \langle f,\, g \rangle = \int_\mathbb{R} f(x)g(x)\, d \mu_0 + \sum_{k=1}^m \int _\mathbb{R} f^{(k)}(x)g^{(k)}(x)\, d \mu_k, \] with \(d \mu_0, \dots, d \mu_m\) positive Borel measures on \(\mathbb{R}\). Properties of the Borel measures \(d \mu_0, \dots, d \mu_m\) as well as several other modifications of the scalar product lead to various kinds of the corresponding orthogonal polynomials. The links between the families of classical polynomials and the Sobolev orthogonal polynomials are discussed. It is shown that the Sobolev orthogonal polynomials can be viewed as generalizations of the classical polynomials. For the scalar product with \(m=1\), the notion of coherent pair of measures \(d \mu_0,\, d \mu_1\) is recalled. Classification results for the coherent pairs as well as various results about generalized \((M,N)\) coherent pairs are summarized. The results for the Sobolev orthogonal polynomials regarding them as solutions of differential equations are reviewed. A summary of results concerning the distribution of zeros and asymptotic properties of these polynomials is also included. The necessity of a further study of the orthogonal expansions and of the multivariate generalizations corresponding to the Sobolev orthogonal polynomials is pointed out.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
41A10 Approximation by polynomials
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
26C10 Real polynomials: location of zeros

Software:

OPQ
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aktaş, R.; Xu, Y., Sobolev orthogonal polynomials on a simplex, Int. Math. Res. Not. IMRN, 13, 3087-3131 (2013) · Zbl 1316.33006
[2] Alfaro, M.; Álvarez de Morales, M.; Rezola, M. L., Orthogonality of the Jacobi polynomials with negative integer parameters, J. Comput. Appl. Math., 145, 379-386 (2002) · Zbl 1002.42016
[3] Alfaro, M.; López Lagomasino, G.; Rezola, M. L., Some properties of zeros of Sobolev-type orthogonal polynomials, J. Comput. Appl. Math., 69, 171-179 (1996) · Zbl 0862.33005
[4] 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) · Zbl 1029.42014
[5] Alfaro, M.; Marcellán, F.; Rezola, M. L.; Ronveaux, A., On orthogonal polynomials of Sobolev type: Algebraic properties and zeros, SIAM J. Math. Anal., 23, 737-757 (1992) · Zbl 0764.33003
[6] Alfaro, M.; Marcellán, F.; Rezola, M. L.; Ronveaux, A., Sobolev type orthogonal polynomials: The nondiagonal case, J. Approx. Theory, 83, 737-757 (1995) · Zbl 0841.42013
[7] Alfaro, M.; Moreno-Balcázar, J. J.; Peña, A.; Rezola, M. L., A new approach to the asymptotics of Sobolev type orthogonal polynomials, J. Approx. Theory, 163, 460-480 (2011) · Zbl 1219.33006
[8] Alfaro, M.; Pérez, T. E.; Piñar, M. A.; Rezola, M. L., Sobolev orthogonal polynomials: the discrete-continuous case, Methods Appl. Anal., 6, 593-616 (1999) · Zbl 0980.42017
[9] Althammer, P., Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation, J. Reine Angew. Math., 211, 192-204 (1962) · Zbl 0108.27204
[10] Álvarez de Morales, M.; Pérez, T. E.; Piñar, M. A., Sobolev orthogonality for the Gegenbauer polynomials \(\{C_n^{(- N + 1 / 2)} \}_{n \geq 0}\), J. Comput. Appl. Math., 100, 111-120 (1998) · Zbl 0931.33008
[11] Álvarez-Nodarse, R.; Moreno-Balcázar, J. J., Asymptotic properties of generalized Laguerre orthogonal polynomials, Indag. Math. (N.S.), 15, 151-165 (2004) · Zbl 1064.41022
[12] Atkinson, K.; Hansen, O., Solving the nonlinear Poisson equation on the unit disk, J. Integral Equations Appl., 17, 223-241 (2005) · Zbl 1096.65119
[13] Bavinck, H.; Meijer, H. G., Orthogonal polynomials with respect to a symmetric inner product involving derivatives, Appl. Anal., 33, 103-117 (1989) · Zbl 0648.33007
[14] Bavinck, H.; Meijer, H. G., On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations, Indag. Math. (N.S.), 1, 7-14 (1990) · Zbl 0704.42023
[15] Berti, A. C.; Bracciali, C. F.; Sri Ranga, A., Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials, Numer. Algorithms, 34, 203-216 (2003) · Zbl 1046.42015
[16] Berti, A. C.; Sri Ranga, A., Companion orthogonal polynomials: some applications, Appl. Numer. Math., 39, 127-149 (2001) · Zbl 1003.42014
[17] Bochner, S., Über Sturm-Liouvillesche polynomsysteme, Math. Z., 89, 730-736 (1929) · JFM 55.0260.01
[18] Boyd, J. P., Chebyshev and Fourier Spectral Methods (2001), Dover: Dover New York · Zbl 0987.65122
[19] Bracciali, C. F.; Delgado, A. M.; Fernández, L.; Pérez, T. E.; Piñar, M. A., New steps on Sobolev orthogonality in two variables, J. Comput. Appl. Math., 235, 916-926 (2010) · Zbl 1200.42011
[20] Brenner, J., Über eine Erweiterung des Orthogonaltäts bei Polynomen, (Alexits, G.; Stechkin, S. B., Constructive Theory of Functions (1972), Akadémiai Kiadó: Akadémiai Kiadó Budapest), 77-83
[21] Cachafeiro, A.; Marcellán, F.; Moreno-Balcázar, J. J., On asymptotic properties of Freud Sobolev orthogonal polynomials, J. Approx. Theory, 125, 26-41 (2003) · Zbl 1043.33005
[22] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods. Fundamentals in Single Domains. Scientific Computation (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1093.76002
[23] Canuto, C.; Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38, 67-86 (1982) · Zbl 0567.41008
[24] Castro, M.; Durán, A. J., Boundedness properties for Sobolev inner products, J. Approx. Theory, 122, 97-111 (2003) · Zbl 1016.42013
[25] Cohen, E. A., Zero distribution and behavior of orthogonal polynomials in the Sobolev space \(W^{1, 2} [- 1, 1]\), SIAM J. Math. Anal., 6, 105-116 (1975) · Zbl 0272.42013
[26] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008
[27] Dai, F.; Xu, Y., (Approximation Theory and Harmonic Analysis on Spheres and Balls. Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer Monographs in Mathematics (2013), Springer: Springer Berlin) · Zbl 1275.42001
[28] de Bruin, M. G., A tool for locating zeros of orthogonal polynomials in Sobolev inner product spaces, J. Comput. Appl. Math., 49, 27-35 (1993) · Zbl 0792.42010
[29] de Bruin, M. G.; Meijer, H. G., Zeros of orthogonal polynomials in a non-discrete Sobolev space, Ann. Numer. Math., 2, 233-246 (1995) · Zbl 0833.33009
[30] de Jesus, M. N.; Marcellán, F.; Petronilho, J.; Pinzón, N., \((M, N)\)-coherent pairs of order \((m, k)\) and Sobolev orthogonal polynomials, J. Comput. Appl. Math., 256, 16-35 (2014) · Zbl 1321.33015
[31] de Jesus, M. N.; Petronilho, J., On linearly related sequences of derivatives of orthogonal polynomials, J. Math. Anal. Appl., 347, 482-492 (2008) · Zbl 1160.42011
[32] de Jesus, M. N.; Petronilho, J., Sobolev orthogonal polynomials and \((M, N)\) coherent pairs of measures, J. Comput. Appl. Math., 237, 83-101 (2013) · Zbl 1258.42024
[33] 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) · Zbl 1205.33020
[34] Delgado, A. M.; Marcellán, F., Companion linear functionals and Sobolev inner products: A case study, Methods Appl. Anal., 11, 237-266 (2004) · Zbl 1087.42020
[35] Delgado, A. M.; Pérez, T. E.; Piñar, M. A., Sobolev-type orthogonal polynomials on the unit ball, J. Approx. Theory, 170, 94-106 (2013) · Zbl 1285.42021
[36] Derevyagin, M.; Marcellán, F., A note on the Geronimus transformation and Sobolev orthogonal polynomials, Numer. Algorithms., 67, 271-287 (2014) · Zbl 1306.42043
[37] Díaz Mendoza, C.; Orive, R.; Pijeira-Cabrera, H., Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights, J. Comput. Appl. Math., 233, 691-698 (2009) · Zbl 1177.42020
[38] Dunkl, C. F.; Xu, Y., (Orthogonal Polynomials of Several Variables. Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 155 (2014), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1317.33001
[39] Durán, A. J., A generalization of Favard’s theorem for polynomials satisfying a recurrence relation, J. Approx. Theory, 74, 83-109 (1993) · Zbl 0789.41017
[40] Durán, A. J.; de la Iglesia, M. D., Differential equations for discrete Laguerre-Sobolev orthogonal polynomials, J. Approx. Theory (2014), in press arXiv:1309.6259
[41] Durán, A. J.; Van Assche, W., Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl., 219, 261-280 (1995) · Zbl 0827.15027
[42] Evans, W. D.; Littlejohn, L. L.; Marcellán, F.; Markett, C.; Ronveaux, A., On recurrence relations for Sobolev orthogonal polynomials, SIAM J. Math. Anal., 26, 446-467 (1995) · Zbl 0824.33006
[43] Fernández, L.; Marcellán, F.; Pérez, T. E.; Piñar, M. A.; Xu, Y., Sobolev orthogonal polynomials on product domain, J. Comput. Appl. Math. (2015), in press arXiv:1406.0762
[44] Gautschi, W., Orthogonal Polynomials: Computation and Approximation (2004), Oxford Univ. Press · Zbl 1130.42300
[45] Gautschi, W.; Kuijlaars, A. B.J., Zeros and critical points of Sobolev orthogonal polynomials, J. Approx. Theory, 91, 117-137 (1997) · Zbl 0897.42014
[46] Gautschi, W.; Zhang, M., Computing orthogonal polynomials in Sobolev spaces, Numer. Math., 71, 159-183 (1995) · Zbl 0830.65012
[47] Geronimo, J. S.; Lubinsky, D. S.; Marcellán, F., Asymptotic for Sobolev orthogonal polynomials for exponential weights, Constr. Approx., 22, 309-346 (2005) · Zbl 1105.42016
[48] Gröbner, W., Orthogonale Polynomsysteme, die Gleichzeitig mit \(f(x)\) auch deren Ableitung \(f^\prime(x)\) approximieren, (Collatz, L., Funktionalanalysis, Approximationstheorie, Numerische Mathematik. Funktionalanalysis, Approximationstheorie, Numerische Mathematik, ISNM, vol. 7 (1967), Birkhäuser: Birkhäuser Basel), 24-32 · Zbl 0188.14001
[49] Guo, B.; Shen, J.; Wang, L., Generalized Jacobi polynomials/functions and applications to spectral methods, Appl. Numer. Math., 59, 1011-1028 (2009) · Zbl 1171.33006
[50] Guo, B.; Wang, L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 128, 1-41 (2004) · Zbl 1057.41003
[51] Hesthaven, J. S.; Gottlieb, S.; Gottlieb, D., (Spectral Methods for Time-dependent Problems. Spectral Methods for Time-dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21 (2007), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 1111.65093
[52] Iliev, P., Krall-Jacobi commutative algebras of partial differential operators, J. Math. Pures Appl. (9), 96, 446-461 (2011) · Zbl 1237.13050
[53] Iliev, P., Krall-Laguerre commutative algebras of ordinary differential operators, Ann. Mat. Pura Appl., 192, 203-224 (2013) · Zbl 1273.33009
[54] 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) · Zbl 0734.42016
[55] Iserles, A.; Sanz-Serna, J. M.; Koch, P. E.; Norsett, S. P., Orthogonality and approximation in a Sobolev space, (Algorithms for Approximation, II (Shrivenham, 1988) (1990), Chapman and Hall: Chapman and Hall London), 117-124 · Zbl 0749.41030
[56] Kim, D. H.; Kwon, K. H.; Marcellán, F.; Yoon, G. J., Sobolev orthogonality and coherent pairs of moment functionals: An inverse problem, Internat. Math. J., 2, 877-888 (2002) · Zbl 1275.42041
[57] Koekoek, R., Generalizations of Laguerre polynomials, J. Math. Anal. Appl., 153, 576-590 (1990) · Zbl 0737.33004
[58] Koekoek, R., The search for differential equations for certain sets of orthogonal polynomials, J. Comput. Appl. Math., 49, 111-119 (1993) · Zbl 0796.34013
[59] Koekoek, J.; Koekoek, R.; Bavinck, H., On differential equations for Sobolev-type Laguerre polynomials, Trans. Amer. Math. Soc., 350, 347-393 (1998) · Zbl 0886.33006
[60] Koekoek, R.; Meijer, H. G., A generalization of Laguerre polynomials, SIAM J. Math. Anal., 24, 768-782 (1993) · Zbl 0780.33007
[61] Kopotun, K., A Note on simultaneous approximation in \(L_p [- 1, 1](1 \leq p < \infty)\), Analysis, 15, 151-158 (1995) · Zbl 0873.41007
[62] Krall, A. M., Orthogonal polynomials satisfying fourth order differential equations, Proc. Roy. Soc. Edinburg, Sect. A, 87, 271-288 (1980/81) · Zbl 0453.33006
[63] Kwon, K. H.; Lee, J. H.; Marcellán, F., Generalized coherent pairs, J. Math. Anal. Appl., 253, 482-514 (2001) · Zbl 0967.33005
[64] Kwon, K. H.; Littlejohn, L. L., The orthogonality of the Laguerre polynomials \(\{L_n^{(- k)}(x) \}\) for positive integers \(k\), Ann. Numer. Anal., 2, 289-303 (1995) · Zbl 0831.33003
[65] Kwon, K. H.; Littlejohn, L. L., Sobolev orthogonal polynomials and second-order differential equations, Rocky Mountain J. Math., 28, 547-594 (1998) · Zbl 0930.33004
[66] Lee, J. K.; Littlejohn, L. L., Sobolev orthogonal polynomials in two variables and second order partial differential equations, J. Math. Anal. Appl., 322, 1001-1017 (2006) · Zbl 1105.33012
[67] Lewis, D. C., Polynomial least square approximations, Amer. J. Math., 69, 273-278 (1947) · Zbl 0033.35603
[68] Li, H.; Shen, J., Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle, Math. Comp., 79, 1621-1646 (2010) · Zbl 1197.65176
[69] Li, H.; Xu, Y., Spectral approximation on the unit ball, SIAM J. Numer. Anal. (2014), in press. arXiv:1310.2283 · Zbl 1315.41002
[70] López Lagomasino, G.; Marcellán, F.; Van Assche, W., Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx., 11, 107-137 (1995) · Zbl 0840.42017
[71] López Lagomasino, G.; Pijeira, H., \(N\) th root asymptotics of Sobolev orthogonal polynomials, J. Approx. Theory, 99, 30-43 (1999) · Zbl 0949.42020
[72] Marcellán, F.; Alfaro, M.; Rezola, M. L., Orthogonal polynomials on Sobolev spaces: old and new directions, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications, VII SPOA, Granada, 1991. Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications, VII SPOA, Granada, 1991, J. Comput. Appl. Math., 48, 113-131 (1993) · Zbl 0790.42015
[73] Marcellán, F.; Martínez-Finkelshtein, A.; Moreno-Balcázar, J. J., \(k\)-Coherence of measures with non-classical weights, (Margarita Mathematica en Memoria de José Javier Guadalupe Hernández (2001), Servicio de Publicaciones: Servicio de Publicaciones Universidad de la Rioja, Logroño, Spain), 77-83 · Zbl 1253.42022
[74] Marcellán, F.; Moreno-Balcázar, J. J., Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports, Acta Appl. Math., 94, 163-192 (2006) · Zbl 1137.42312
[75] Marcellán, F.; Pinzón, N., Higher order coherent pairs, Acta Appl. Math., 121, 105-135 (2012) · Zbl 1262.42009
[76] Marcellán, F.; Pérez, T. E.; Piñar, M. A., On zeros of Sobolev type orthogonal polynomials, Rend. Mat. Appl. (7), 12, 455-473 (1992) · Zbl 0768.33008
[77] Marcellán, F.; Pérez, T. E.; Piñar, M. A., Orthogonal polynomials on weighted Sobolev spaces: the semiclassical case, Ann. Numer. Math., 2, 93-122 (1995) · Zbl 0835.33005
[78] Marcellán, F.; Pérez, T. E.; Piñar, M. A., Laguerre-Sobolev orthogonal polynomials, J. Comput. Appl. Math., 71, 245-265 (1996) · Zbl 0855.33015
[79] Marcellán, F.; Petronilho, J. C., Orthogonal polynomials and coherent pairs: The classical case, Indag. Math. (N.S.), 6, 287-307 (1995) · Zbl 0843.42010
[80] Marcellán, F.; Ronveaux, A., On a class of polynomials orthogonal with respect to a discrete Sobolev inner product, Indag. Math. (N.S.), 1, 451-464 (1990) · Zbl 0732.42016
[81] Marcellán, F.; Van Assche, W., Relative asymptotics for orthogonal polynomials with a Sobolev inner product, J. Approx. Theory, 72, 193-209 (1993) · Zbl 0771.42014
[82] Marcellán, F.; Zejnullahu, R. Xh.; Xh Fejzullahu, B.; Huertas, E., On orthogonal polynomials with respect to certain discrete Sobolev inner product, Pacific J. Math., 257, 167-188 (2012) · Zbl 1259.33023
[83] Maroni, P., Une théorie algébrique des polynômes orthogonaux: Application aux polynômes, orthogonaux semi-classiques, (Brezinski, C.; Gori, L.; Ronveaux, A., Orthogonal Polynomials and Their Applications. Orthogonal Polynomials and Their Applications, IMACS Ann. Comput. Appl. Math., vol. 9 (1991), Baltzer: Baltzer Basel), 95-130 · Zbl 0944.33500
[84] Martínez-Finkelshtein, A., Asymptotic properties of Sobolev orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications, Seville, 1997. Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications, Seville, 1997, J. Comput. Appl. Math., 99, 491-510 (1998) · Zbl 0933.42013
[85] Martínez-Finkelshtein, A., Bernstein-Szegő’s theorem for Sobolev orthogonal polynomials, Constr. Approx., 16, 73-84 (2000) · Zbl 0951.42014
[86] Martínez-Finkelshtein, A., Analytic aspects of Sobolev orthogonal polynomials revisited, Numerical Analysis 2000, Vol. V, Quadrature and Orthogonal Polynomials. Numerical Analysis 2000, Vol. V, Quadrature and Orthogonal Polynomials, J. Comput. Appl. Math., 127, 255-266 (2001) · Zbl 0971.33004
[87] Martínez-Finkelshtein, A.; Moreno-Balcázar, J. J.; Pérez, T. E.; Piñar, M. A., Asymptotics of Sobolev orthogonal polynomials for coherent pairs, J. Approx. Theory, 92, 280-293 (1998) · Zbl 0898.42006
[88] Martínez-Finkelshtein, A.; Pijeira-Cabrera, H., Strong asymptotics for Sobolev orthogonal polynomials, J. Anal. Math., 78, 143-156 (1999) · Zbl 0937.42011
[89] Meijer, H. G., Coherent pairs and zeros of Sobolev-type orthogonal polynomials, Indag. Math. (N.S.), 4, 163-176 (1993) · Zbl 0784.33004
[90] Meijer, H. G., Sobolev orthogonal polynomials with a small number of real zeros, J. Approx. Theory, 77, 305-313 (1994) · Zbl 0806.42015
[91] Meijer, H. G., A short history of orthogonal polynomials in a Sobolev space I. The non-discrete case, 31st Dutch Mathematical Conference, Groningen, 1995. 31st Dutch Mathematical Conference, Groningen, 1995, Nieuw Arch. Wiskd. (4), 14, 93-112 (1996) · Zbl 0862.33001
[92] Meijer, H. G., Determination of all coherent pairs of functionals, J. Approx. Theory, 89, 321-343 (1997) · Zbl 0880.42012
[93] Meijer, H. G.; de Bruin, M. G., Zeros of Sobolev orthogonal polynomials following from coherent pairs, J. Comput. Appl. Math., 139, 253-274 (2002) · Zbl 1005.42015
[94] Moreno-Balcázar, J. J., A note on the zeros of Freud-Sobolev orthogonal polynomials, J. Comput. Appl. Math., 207, 338-344 (2007) · Zbl 1120.33009
[95] Pérez, T. E.; Piñar, M. A., On Sobolev orthogonality for the generalized Laguerre polynomials, J. Approx. Theory, 86, 278-285 (1996) · Zbl 0864.33009
[96] Pérez, T. E.; Piñar, M. A.; Xu, Y., Weighted Sobolev orthogonal polynomials on the unit ball, J. Approx. Theory, 171, 84-104 (2013) · Zbl 1281.42026
[97] Piñar, M.; Xu, Y., Orthogonal polynomials and partial differential equations on the unit ball, Proc. Amer. Math. Soc., 137, 2979-2987 (2009) · Zbl 1178.33009
[98] Schäfke, F. W., Zu den Orthogonalpolynomen von Althammer, J. Reine Angew. Math., 252, 195-199 (1972) · Zbl 0226.33012
[99] Schäfke, F. W.; Wolf, G., Einfache verallgemeinerte klassische Orthogonal polynome, J. Reine Angew. Math., 262/263, 339-355 (1973) · Zbl 0272.33018
[100] Shen, J.; Wang, L.; Li, H., A triangular spectral element method using fully tensorial rational basis functions, SIAM J. Numer. Anal., 47, 1619-1650 (2009) · Zbl 1197.65187
[101] Szegő, G., (Orthogonal Polynomials. Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23 (1975), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · JFM 65.0278.03
[102] Xu, Y., A family of Sobolev orthogonal polynomials on the unit ball, J. Approx. Theory, 138, 232-241 (2006) · Zbl 1092.42016
[103] Xu, Y., Sobolev orthogonal polynomials defined via gradient on the unit ball, J. Approx. Theory, 152, 52-65 (2008) · Zbl 1197.42016
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.