Constructive Approximation

, Volume 8, Issue 2, pp 187–201

Polynomial approximation inLp (0<p<1)

  • Ronald A. DeVore
  • Dany Leviatan
  • Xiang Ming Yu
Article

Abstract

We prove that forfLp, 0<p<1, andk a positive integer, there exists an algebraic polynomialPn of degree ≤n such that
$$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$
whereωkϕ(f,t)p is the Ditzian-Totik modulus of smoothness off inLp, andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialPn can also be taken to be nondecreasing.

AMS classification

41A25 41A20 

Key words and phrases

Degree of approximation Monotone approximation Polynomials 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-L]L. G. Brown, B. J. Lucier (to appear):Best approximations in L 1 are near best in L p, 0<p<1, Proc. Amer. Math. Soc.Google Scholar
  2. [D-L]R. DeVore andG. G. Lorentz (1992) Constructive Approximation. New York: Springer-Verlag.Google Scholar
  3. [D-P]R. DeVore, V. Popov (1987):Interpolation of Besov spaces. Trans. Amer. Math. Soc.,305: 397–414.Google Scholar
  4. [D-Y]R. A. DeVore, X. M. Yu (1985):Pointwise estimates for monotone polynomial approximation. Constr. Approx.,1:323–331.Google Scholar
  5. [D-T]Z. Ditzian, V. Totik (1987): Moduli of Smoothness. Series in Computational Mathematics. New York: Springer-Verlag.Google Scholar
  6. [K]L. B. Khodak (1981):Approximation of functions by algebraic polynomials in the metric L p for 0<p<1. Mat. Zametki,30:649–655.Google Scholar
  7. [L]D. Leviatan (1988):Monotone and comontone approximation revisited, J. Approx. Theory,53:1–16.Google Scholar
  8. [L-Y]D. Leviatan, X. M. Yu (1991):Shape preserving approximation by polynomial in L p. Preprint.Google Scholar
  9. [Lo]G. G. Lorentz (1966): Approximation of Functions. New York: Holt, Rinehart and Winston.Google Scholar
  10. [P-P]P. Petrushev, V. Popov (1987): Rational Approximation of Real Functions. Cambridge: Cambridge University Press.Google Scholar
  11. [S]A. S. Shvedov (1979):Orders of coapproximation. English Trans. Math. Notes,25:57–63; Mat. Zametki,25:107–117.Google Scholar
  12. [S-K-O]E. A. Storozhenko, V. G. Krotov, P. Oswald (1975):Jackson-type direct and converse theorems in L p,0<p<1, spaces. Mat. Sbornik,98:395–415.Google Scholar
  13. [Y]X. M. Yu (1987):Monotone polynomial approximation in L p spaces. Acta Math. Sinica (New Series),3:315–326.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Ronald A. DeVore
    • 1
  • Dany Leviatan
    • 2
  • Xiang Ming Yu
    • 1
  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations