Stability properties of disk polynomials

Abstract

Disk polynomials form a basis of orthogonal polynomials on the disk corresponding to the radial weight \({\alpha +1 \over \pi }(1-r^{2})^{\alpha }\). In this paper, the stability properties of disk polynomials are analyzed. A conditioning associated with the representation of the least squares approximation with respect to this basis is introduced and bounded. Among all disk polynomials, the least bounds are obtained for Zernike polynomials corresponding to α = 0.

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References

  1. 1.

    Askey, R.: Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces. Bull. Amer. Math. Soc. 74, 301–304 (1968)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Beals, R., Wong, R.: Special Functions and Orthogonal Polynomials Cambridge Studies in Advanced Mathematics, vol. 153. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  3. 3.

    Briani, M., Sommariva, A., Vianello, M.: Computing Fekete and Lebesgue points: simplex, square, disk. J. Comput. Appl. Math. 236, 2477–2486 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Carnicer, J.M., Godés, C.: Interpolation on the disk. Numer. Algor. 66, 1–16 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Carnicer, J.M., Khiar, Y., Peña, J.M.: Optimal stability of the Lagrange formula and conditioning of the Newton formula. J. Approx. Theory 238, 52–66 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Carnicer, J.M., Khiar, Y., Peña, J.M.: Conditioning of polynomial Fourier sums, Calcolo 56. Art. 24, 23 (2019)

    MATH  Google Scholar 

  7. 7.

    Cuyt, A., Yaman, I., Ibrahimoglu, B.A., Benouahmane, B.: Radial orthogonality and Lebesgue constants on the disk. Numer. Algor. 61, 291–313 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, Second Edition Encyclopedia of Mathematics and Its Applications, vol. 155. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  9. 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)

  10. 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)

  11. 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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Logan, B.F., Shepp, L.A.: Optimal reconstruction of a function from its projections. Duke. Math. J. 42, 645–659 (1975)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Lyche, T., Peña, J.M.: Optimally stable multivariate bases. Adv. Comput. Math. 20, 149–159 (2004)

    MathSciNet  Article  Google Scholar 

  14. 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)

  15. 15.

    Pap, M., Schipp, F.: Discrete orthogonality of Zernike functions. Mathematica Pannonica 16, 137–144 (2005)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Szegő, G.: Orthogonal Polynomials Colloquium Publ., vol. 23. American Mathematical Society, Providence, Rhode Island (2003)

    Google Scholar 

  17. 17.

    Vasil, G.M., Burns, K.J., Lecoanet, D., Olver, S., Brown, B.P., Oishi, J.S.: Tensor calculus in polar coordinates using Jacobi polynomials. J. Comput. Phys. 325, 53–73 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Waldron, S.: Orthogonal polynomials on the disc. J. Approx. Theory 150, 117–131 (2008)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Waldron, S.: Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight. Constr. Approx. 30, 33–52 (2009)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Wünsche, A.: Generalized Zernike or disc polynomials. J. Comput. Appl. Math. 174, 135–163 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Xu, Y.: Representation of reproducing kernels and the Lebesgue constants on the ball. J. Approx. Theory 112, 295–310 (2001)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Zernike, F.: Beugungstheorie des Schneidensverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. Physica 1, 689–704 (1934)

    Article  Google Scholar 

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Funding

This work has been partially supported by the PGC2018-096321-B-I00 Spanish Research Grant, by Gobierno de Aragó n E41_17R and Feder 2014-2020 “Construyendo Europa desde Aragón”.

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Correspondence to E. Mainar.

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Carnicer, J.M., Mainar, E. & Peña, J.M. Stability properties of disk polynomials. Numer Algor (2020). https://doi.org/10.1007/s11075-020-00960-3

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Keywords

  • Orthogonal polynomials
  • Disk polynomials
  • Zernike polynomials
  • Lebesgue constant
  • conditioning

Mathematics Subject Classification (2010)

  • 41A10
  • 41A63
  • 33C50
  • 65F35