Constructive Approximation

, Volume 46, Issue 1, pp 171–197 | Cite as

Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the Unit Disk

  • Leonardo E. Figueroa


We study approximation properties of weighted \(L^2\)-orthogonal projectors onto the space of polynomials of degree less than or equal to N on the unit disk where the weight is of the generalized Gegenbauer form \(x \mapsto (1-\left|x\right|^2)^\alpha \). The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted \(L^2\) norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and nonuniformly weighted Sobolev spaces involved, connection coefficients between Zernike polynomials, an inverse inequality, and relations between the Fourier–Zernike expansions of a function and its derivatives.


Zernike polynomials Connection coefficients Orthogonal projection Weighted Sobolev space 

Mathematics Subject Classification

41A25 41A10 42C10 46E35 



We thank the anonymous referees for their helpful and constructive suggestions.


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Aharmim, B., Amal, E.H., Fouzia, E.W., Ghanmi, A.: Generalized Zernike polynomials: operational formulae and generating functions. Integr. Transf. Spec. F. 26(6), 395–410 (2015). doi: 10.1080/10652469.2015.1012510 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  4. 4.
    Bhatia, A.B., Wolf, E.: On the circle polynomials of Zernike and related orthogonal sets. Proc. Camb. Philos. Soc. 50, 40–48 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boyd, J.P., Yu, F.: Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions. J. Comput. Phys. 230(4), 1408–1438 (2011). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)zbMATHGoogle Scholar
  7. 7.
    Cação, I., Morais, J.: An orthogonal set of weighted quaternionic Zernike spherical functions. In: Computational Science and Its Applications—ICCSA 2014. Part I. Lecture Notes in Computer Science, vol. 8579, pp. 103–116. Springer, Cham (2014). doi: 10.1007/978-3-319-09144-0_8
  8. 8.
    Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38(157), 67–86 (1982). doi: 10.2307/2007465 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dai, F., Xu, Y.: Moduli of smoothness and approximation on the unit sphere and the unit ball. Adv. Math. 224(4), 1233–1310 (2010). doi: 10.1016/j.aim.2010.01.001 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dai, F., Xu, Y.: Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball. J. Approx. Theory 163(10), 1400–1418 (2011). doi: 10.1016/j.jat.2011.05.001 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (2014). doi: 10.1017/CBO9781107786134
  14. 14.
    Figueroa, L.E.: Deterministic simulation of multi-beaded models of dilute polymer solutions. Ph.D. thesis, University of Oxford (2011).
  15. 15.
    Figueroa, L.E.: Orthogonal polynomial projection error measured in Sobolev norms in the unit disk. Tech. rep., (2015). arXiv:1503.04485v2
  16. 16.
    Girault, V., Raviart, P.A.: Finite element methods for Navier–Stokes equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986). doi: 10.1007/978-3-642-61623-5
  17. 17.
    Glaeske, H.J.: On Zernicke [Zernike] transforms in spaces of distributions. Integr. Transf. Spec. F. 4(3), 221–234 (1996). doi: 10.1080/10652469608819109 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guo, B.Y.: Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations. SIAM J. Numer. Anal. 37(2), 621–645 (2000). doi: 10.1137/S0036142998342161 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Janssen, A., Dirksen, P.: Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform. J. Eur. Opt. Soc. Rapid Public. 2, 07012 (2007). doi: 10.2971/jeos.2007.07012
  20. 20.
    Janssen, A.J.E.M.: Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials. J. Opt. Soc. Am. A 31(7), 1604–1613 (2014). doi: 10.1364/JOSAA.31.001604 CrossRefGoogle Scholar
  21. 21.
    Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 435–495. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. Academic Press, New York (1975)Google Scholar
  22. 22.
    Koornwinder, T.H.: The addition formula for Jacobi polynomials II. The Laplace type integral and the product formula. Tech. Rep. 133, Math. Centrum Afd. Toegepaste Wisk. (1972)Google Scholar
  23. 23.
    Kufner, A.: Weighted Sobolev Spaces. Wiley, New York (1985)zbMATHGoogle Scholar
  24. 24.
    Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carol. 25(3), 537–554 (1984)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Li, H., Xu, Y.: Spectral approximation on the unit ball. SIAM J. Numer. Anal. 52(6), 2647–2675 (2014). doi: 10.1137/130940591 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Noll, R.J.: Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am. 66(3), 207–211 (1976). doi: 10.1364/JOSA.66.000207 CrossRefGoogle Scholar
  27. 27.
    Olver, F.W.J.: Asymptotics and special functions. AKP Classics. A K Peters, Ltd., Wellesley, MA (1997)Google Scholar
  28. 28.
    Opic, B., Gurka, P.: Continuous and compact imbeddings of weighted Sobolev spaces. II. Czechoslov. Math. J. 39(1), 78–94 (1989)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sheppard, C.J.R., Campbell, S., Hirschhorn, M.D.: Zernike expansion of separable functions of cartesian coordinates. Appl. Opt. 43(20), 3963–3966 (2004). doi: 10.1364/AO.43.003963 CrossRefGoogle Scholar
  30. 30.
    Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)zbMATHGoogle Scholar
  31. 31.
    Tartar, L.: An introduction to Sobolev spaces and interpolation spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3. Springer, Berlin (2007)Google Scholar
  32. 32.
    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). doi: 10.1016/ MathSciNetCrossRefGoogle Scholar
  33. 33.
    Waldron, S.: Orthogonal polynomials on the disc. J. Approx. Theory 150(2), 117–131 (2008). doi: 10.1016/j.jat.2007.05.001 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wünsche, A.: Generalized Zernike or disc polynomials. J. Comput. Appl. Math. 174(1), 135–163 (2005). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Xu, Y.: Weighted approximation of functions on the unit sphere. Constr. Approx. 21(1), 1–28 (2005). doi: 10.1007/s00365-003-0542-5 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CI²MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile

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