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Stability properties of disk polynomials. (English) Zbl 1464.41002

Zernike polynomials are used in optics to represent wavefronts and describe optical aberrations. Disk polynomials are generalizations of Zernike polynomials for radial weights. In this paper the stability properties of disk polynomials have been analyzed. A conditioning associated with the representation of the least squares approximation with respect to the bases formed by disk polynomials corresponding to radial weight have been introduced. It has been shown that among all disk polynomials, the least bounds are obtained for Zernike polynomials corresponding to the radial weight \(\frac{\alpha+1}{\pi}(1-r^{2})^{\alpha}\) for \(\alpha=0\).

MSC:

41A10 Approximation by polynomials
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
41A63 Multidimensional problems
65F35 Numerical computation of matrix norms, conditioning, scaling

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[1] Askey, R., Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces, Bull. Amer. Math. Soc., 74, 301-304 (1968) · Zbl 0167.35003 · doi:10.1090/S0002-9904-1968-11931-7
[2] Beals, R.; Wong, R., Special Functions and Orthogonal Polynomials Cambridge Studies in Advanced Mathematics, vol. 153 (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1365.33001 · doi:10.1017/CBO9781316227381
[3] Briani, M.; Sommariva, A.; Vianello, M., Computing Fekete and Lebesgue points: simplex, square, disk, J. Comput. Appl. Math., 236, 2477-2486 (2012) · Zbl 1242.41002 · doi:10.1016/j.cam.2011.12.006
[4] Carnicer, JM; Godés, C., Interpolation on the disk, Numer. Algor., 66, 1-16 (2014) · Zbl 1295.41003 · doi:10.1007/s11075-013-9720-0
[5] Carnicer, JM; Khiar, Y.; Peña, JM, Optimal stability of the Lagrange formula and conditioning of the Newton formula, J. Approx. Theory, 238, 52-66 (2019) · Zbl 1407.41002 · doi:10.1016/j.jat.2017.07.005
[6] Carnicer, JM; Khiar, Y.; Peña, JM, Conditioning of polynomial Fourier sums, Calcolo 56, Art., 24, 23 (2019) · Zbl 1420.65061
[7] Cuyt, A.; Yaman, I.; Ibrahimoglu, BA; Benouahmane, B., Radial orthogonality and Lebesgue constants on the disk, Numer. Algor., 61, 291-313 (2012) · Zbl 1259.65011 · doi:10.1007/s11075-012-9615-5
[8] Dunkl, CF; Xu, Y., Orthogonal Polynomials of Several Variables, Second Edition Encyclopedia of Mathematics and Its Applications, vol. 155 (2014), Cambridge: Cambridge University Press, Cambridge · Zbl 1317.33001
[9] Koornwinder, T.H.: The Addition Formula for Jacobi Polynomials II. The Laplace Type Integral and the Product Formula, Report TW 133/72, Mathematisch Centrum, Amsterdam, https://staff.fnwi.uva.nl/t.h.koornwinder/art/index.html#1972 (1972) · Zbl 0247.33018
[10] Koornwinder, T.H.: The Addition Formula for Jacobi Polynomials III. Completion of the Proof, Report TW 135/72, Mathematisch Centrum, Amsterdam, https://staff.fnwi.uva.nl/t.h.koornwinder/art/index.html#1972 (1972) · Zbl 0247.33019
[11] Koornwinder, T.; Kostenko, A.; Teschl, G., Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator, Adv. Math., 333, 796-821 (2018) · Zbl 1405.33012 · doi:10.1016/j.aim.2018.05.038
[12] Logan, BF; Shepp, LA, Optimal reconstruction of a function from its projections, Duke. Math. J., 42, 645-659 (1975) · Zbl 0343.41020
[13] Lyche, T.; Peña, JM, Optimally stable multivariate bases, Adv. Comput. Math., 20, 149-159 (2004) · Zbl 1041.65017 · doi:10.1023/A:1025863309959
[14] Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, Department of Commerce, Nationa Institute of Standards and Technology, Washington (2010) · Zbl 1198.00002
[15] Pap, M.; Schipp, F., Discrete orthogonality of Zernike functions, Mathematica Pannonica, 16, 137-144 (2005) · Zbl 1082.42019
[16] Szegő, G., Orthogonal Polynomials Colloquium Publ., vol. 23 (2003), Rhode Island: American Mathematical Society, Providence, Rhode Island
[17] Vasil, GM; Burns, KJ; Lecoanet, D.; Olver, S.; Brown, BP; Oishi, JS, Tensor calculus in polar coordinates using Jacobi polynomials, J. Comput. Phys., 325, 53-73 (2016) · Zbl 1380.65392 · doi:10.1016/j.jcp.2016.08.013
[18] Waldron, S., Orthogonal polynomials on the disc, J. Approx. Theory, 150, 117-131 (2008) · Zbl 1132.33317 · doi:10.1016/j.jat.2007.05.001
[19] Waldron, S., Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight, Constr. Approx., 30, 33-52 (2009) · Zbl 1180.33011 · doi:10.1007/s00365-008-9021-3
[20] Wünsche, A., Generalized Zernike or disc polynomials, J. Comput. Appl. Math., 174, 135-163 (2005) · Zbl 1062.33011 · doi:10.1016/j.cam.2004.04.004
[21] Xu, Y., Representation of reproducing kernels and the Lebesgue constants on the ball, J. Approx. Theory, 112, 295-310 (2001) · Zbl 0988.42016 · doi:10.1006/jath.2001.3597
[22] Zernike, F., Beugungstheorie des Schneidensverfahrens und seiner verbesserten Form, der Phasenkontrastmethode, Physica, 1, 689-704 (1934) · Zbl 0009.28101 · doi:10.1016/S0031-8914(34)80259-5
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