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Sobolev orthogonal polynomials: asymptotics and symbolic computation. (English) Zbl 1510.42036

Summary: The Sobolev polynomials, which are orthogonal with respect to an inner product involving derivatives, are considered. The theory about these nonstandard polynomials has been developed along the last 40 years. The local asymptotics of these polynomials can be described by the Mehler-Heine formulae, which connect the polynomials with the Bessel functions of the first kind. In recent years, the formulae have been computed for discrete Sobolev orthogonal polynomials in several particular cases. We improve various known results by unifying them. Besides, an algorithm to compute these formulae effectively is presented. The algorithm allows to construct a computer program based on \(Mathematica^{\circledR}\) language, where the corresponding Mehler-Heine formulae are automatically obtained. Applications and examples show the efficiency of the approach developed.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
33C47 Other special orthogonal polynomials and functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
65D20 Computation of special functions and constants, construction of tables

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[1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover (1972). · Zbl 0543.33001
[2] M. Alfaro, F. Marcellán, H.G. Meijer and M.L. Rezola, Symmetric orthogonal polynomials for Sobolev-type inner products, J. Math. Anal. Appl. 184, 360-381 (1994). · Zbl 0809.42012
[3] M. Alfaro, J.J. Moreno-Balcázar, A. Peña and M.L. Rezola, Asymptotic formulae for generalized Freud polynomials, J. Math. Anal. Appl. 421, 474-488 (2015). · Zbl 1295.30014
[4] P. Althammer, Eine erweiterung des orthogonalitätsbegriffes bei polynomen und deren anwendung auf die beste approximation, J. Reine Angew. Math. 211, 192-204 (1962). · Zbl 0108.27204
[5] A.I. Aptekarev, Asymptotics of orthogonal polynomials in a neighborhood of the endpoints of the interval of orthogonality, Russian Acad. Sci. Sb. Math. 76, 35-50 (1993). · Zbl 0782.33005
[6] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publish-ers, (1978). · Zbl 0389.33008
[7] B.Xh. Fejzullahu, Mehler-Heine formulas for orthogonal polynomials with respect to the modified Jacobi weight, Proc. Amer. Math. Soc. 142, 2035-2045 (2014). · Zbl 1290.42054
[8] M.I. Ganzburg, Limit theorems of polynomial approximation with exponential weights, Mem. Amer. Math. Soc. 192, Providence (2008). · Zbl 1142.30011
[9] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, (2004). · Zbl 1130.42300
[10] W. Gautschi and A.B.J. Kuijlaars, Zeros and critical points of Sobolev orthogonal polynomials, J. Approx. Theory 91, 117-137 (1997). · Zbl 0897.42014
[11] W. Gröbner, Orthogonale Polynomsysteme, die Gleichzeitig mit f(x) auch deren Ableitung f’(x) approximieren, in: Funktionalanalysis, Approximationstheorie, Numerische Mathematik, L. Col-latz, H. Unger and G. Meinardus (Eds), pp. 24-32, Birkhauser Verlag, (1967). · Zbl 0188.14001
[12] A. Iserles, P.E. Koch, S.P. Nørsett and J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65, 151-175 (1991). · Zbl 0734.42016
[13] A. Iserles, J.M. Sanz-Serna, P.E. Koch and S.P. Nørsett, Orthogonality and approximation in a Sobolev space, in: Algorithms for Approximation, II, pp. 117-124, Chapman and Hall, (1990). · Zbl 0749.41030
[14] W. Koepf, Software for the algorithmic work with orthogonal polynomials and special functions, Electron. Trans. Numer. Anal. 9, 77-101 (1999). · Zbl 0946.33001
[15] W. Koepf, Computer algebra algorithms for orthogonal polynomials and special functions, in: Orthogonal polynomials and special functions (Leuven, 2002), pp. 1-24, Lecture Notes in Math. Springer, (2003). · Zbl 1031.33014
[16] W. Koepf, Computer algebra methods for orthogonal polynomials, in: Difference equations, spe-cial functions and orthogonal polynomials, World Sci. Publ. 325-343 (2007). · Zbl 1127.65015
[17] D.C. Lewis, Polynomial least square approximations, Amer. J. Math. 6, 273-278 (1947). · Zbl 0033.35603
[18] J.F. Mañas-Mañas, F. Marcellán and J.J. Moreno-Balcázar, Asymptotic behavior of varying dis-crete Jacobi-Sobolev orthogonal polynomials, J. Comput. Appl. Math. 300, 341-353 (2016). · Zbl 1334.33029
[19] J.F. Mañas-Mañas, F. Marcellán and J.J. Moreno-Balcázar, Asymptotics for varying discrete Sobolev orthogonal polynomials, Appl. Math. Comput. 314, 65-79 (2017). · Zbl 1426.33034
[20] J.F. Mañas-Mañas, J.J. Moreno-Balcázar and R. Wellman, Eigenvalue problem for discrete Jacobi-Sobolev orthogonal polynomials, Mathematics 8, Art. 182 (2020).
[21] F. Marcellán and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math. 33, 308-352 (2015). · Zbl 1351.33011
[22] P. Nevai, Orthogonal Polynomials, Mem. Amer. Math. Soc. 18, 185 pp. (1979). · Zbl 0405.33009
[23] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Func-tions, Cambridge University Press, (2010). · Zbl 1198.00002
[24] A. Peña and M.L. Rezola, Connection formulas for general discrete Sobolev polynomials: Mehler-Heine asymptotics, Appl. Math. Comput. 261, 216-230 (2015). · Zbl 1410.42026
[25] Y. Ren, X. Yu and Z. Wang, Diagonalized Chebyshev rational spectral methods for second-order el-liptic problems on unbounded domains, Numer. Math. Theor. Meth. Appl. 12, 265-284 (2019). · Zbl 1438.65307
[26] F.W. Schäfke, Zu den Orthogonalpolynomen von Althammer, J. Reine Angew. Math. 252, 195-199 (1972). · Zbl 0226.33012
[27] F.W. Schäfke and G. Wolf, Einfache verallgemeinerte klassische Orthogonalpolynome, J. Reine Angew. Math. 262/263, 339-355 (1973). · Zbl 0272.33018
[28] I.I. Sharapudinov, Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs, Izv. Ross. Akad. Nauk Ser. Mat. 83, 204-226 (2019). (English Translation in: Izv. Math. 83, 391-412 (2019).) · Zbl 1412.42067
[29] I.I. Sharapudinov, Sobolev-orthogonal systems of functions and some of their applications, Us-pekhi Mat. Nauk, 74 , 87-164 (2019). (English Translation in Russian Math. Surveys 74, 659-733 (2019).) · Zbl 1434.42039
[30] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. 23, AMS, (1975).
[31] X. Yu, Z. Wang and H. Li, Jacobi-Sobolev orthogonal polynomials and spectral methods for elliptic boundary value problems, Commun. Appl. Math. Comput. 1, 283-308 (2019). · Zbl 1449.33014
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