Skip to main content
SpringerLink
Log in

Polynomial Approximation on the Unit Ball

  • Chapter
  • First Online:
Approximation Theory and Harmonic Analysis on Spheres and Balls

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 2661 Accesses

Abstract

We study the problem of characterizing the best approximation by polynomials on the unit ball in terms of the smoothness of the function being approximated,similar to what we did on the unit sphere in Chap. 4. There is, however, an essential difference between approximations on the unit ball and those on the unit sphere, which arises from the simple fact that the ball is a domain with boundary, whereas the sphere has no boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bagby, T., Bos, L., Levenberg, N.: Multivariate simultaneous approximation. Constr. Approx. 18, 569–577(2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Butzer, P.L., Jansche, S., Stens, R.L.: Functional analytic methods in the solution of the fundamental theorems on best weighted algebraic approximation. In: Approximation Theory(Memphis, TN, 1991). Lecture Notes in Pure and Applied Mathematics, vol. 138, pp. 151–205. Dekker, New York(1992)

    Google Scholar 

  3. Dai, F.: Multivariate polynomial inequalities with respect to doubling weights and A ∞ weights. J. Funct. Anal. 235, 137–170(2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dai, F., Ditzian, Z.: Littlewood–Paley theory and a sharp Marchaud inequality. Acta Sci. Math.(Szeged) 71, 65–90(2005)

    MathSciNet  MATH  Google Scholar 

  5. Dai, F., Ditzian, Z., Huang, H.W.: Equivalence of measures of smoothness in L p (S d − 1), 1 < p < ∞. Studia Math. 196, 179–205(2010)

    Google Scholar 

  6. Dai, F., Huang, H., Wang, K.: Approximation by Bernstein–Durrmeyer operator on a simplex. Constr. Approx. 31, 289–308(2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dai, F., Xu, Y.: Moduli of smoothness and approximation on the unit sphere and the unit ball. Adv. Math. 224(4), 1233–1310(2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dai, F., Xu, Y.: Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball. J. Approx. Theor. 163, 1400–1418(2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ditzian, Z.: Polynomial approximation and ω φ r (f, t) twenty years later. Surv. Approx. Theor. 3, 106–151(2007)

    MathSciNet  MATH  Google Scholar 

  10. Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, Berlin(1987)

    Book  MATH  Google Scholar 

  11. Hu, Y., Liu, Y.: On equivalence of moduli of smoothness of polynomials in L p , 0 < p ≤ ∞. J. Approx. Theor. 136, 182–197(2005)

    Google Scholar 

  12. Ivanov, K.: A characterization of weighted Peetre K-functionals. J. Approx. Theor. 56, 185–211(1989)

    Article  MATH  Google Scholar 

  13. Lorentz, G.G.: Approximation of Functions. Holt, Rinehart and Winston, New York(1966)

    MATH  Google Scholar 

  14. Nikolskii, S.M.: On the best approximation of functions satisfying Lipschitz’s conditions by polynomials(Russian). Bull. Acad. Sci. URSS. Ser. Math. [Izvestia Akad. Nauk SSSR] 10, 295–322(1946)

    Google Scholar 

  15. Potapov, M.K., Kazimirov, G.N.: On the approximation of functions that have a given order of k-th generalized modulus of smoothness by algebraic polynomials(Russian). Mat. Zametki 63(3), 425–436(1998). Translation in Math. Notes 63(3–4), 374–383(1998)

    Google Scholar 

  16. Ragozin, D.L.: Constructive polynomial approximation on spheres and projective spaces. Trans. Am. Math. Soc. 162, 157–170(1971)

    MathSciNet  Google Scholar 

  17. Timan, A.F.: A strengthening of Jackson’s theorem on the best approximation of continuous functions by polynomials on a finite segment of the real axis(Russian). Doklady Akad. Nauk SSSR(N.S.) 78, 17–20(1951)

    Google Scholar 

  18. Timan, A.F.: Theory of approximation of functions of a real variable. Translated from the Russian by J. Berry. Translation edited and with a preface by J. Cossar. Reprint of the 1963 English translation. Dover Publications, Mineola, New York(1994)

    Google Scholar 

  19. Xu, Y.: Weighted approximation of functions on the unit sphere. Constr. Approx. 21, 1–28(2005)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Feng Dai

  2. Department of Mathematics, University of Oregon, Eugene, OR, USA

    Yuan Xu

Authors
  1. Feng Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Dai, F., Xu, Y. (2013). Polynomial Approximation on the Unit Ball. In: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6660-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation