Advertisement

Israel Journal of Mathematics

, Volume 75, Issue 2–3, pp 167–191 | Cite as

Orthonormal polynomial bases in function spaces

  • P. Wojtaszczyk
  • K. Woźniakowski
Article

Abstract

We construct polynomial orthonormal bases in various function spaces. Our bases have linear order of growth of degrees of polynomials. We show that this order is optimal.

Keywords

Fourier Coefficient Trigonometric Polynomial Polynomial Basis Unconditional Basis ORTHONORMAL Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Boč]
    S. V. Bočkariov,Construction using Fejer kernels of interpolating dyadic basis in the space of continuous functions, Trudy Steklov Inst.172 (1985), 29–59 (in Russian).Google Scholar
  2. [Boč2]
    S. V. Bočkariov,Construction of polynomial bases in finite dimensional spaces of functions analytic in the unit disc, Trudy Steklov Inst.164 (1983), 49–74 (in Russian).Google Scholar
  3. [Boč3]
    S. V. Bočkariov,Conjugate Franklin system — a basis in the space of continuous functions, Dokl. Akad. Nauk285 (1985), 521–526 (in Russian).Google Scholar
  4. [Bou]
    J. Bourgain,Homogeneous polynomials on the ball and polynomial bases, Isr. J. Math.68 (1989), 327–347.MATHCrossRefMathSciNetGoogle Scholar
  5. [Čan1]
    Z. A. Čanturija,On orthogonal polynomial bases in the spaces C and L, Analysis Math5 (1) (1979), 9–17.CrossRefGoogle Scholar
  6. [Čan2]
    Z. A. Chanturia,On unconditional polynomial bases of the space L p, Studia Math.71 (1981), 163–168.MATHMathSciNetGoogle Scholar
  7. [C-W]
    R. R. Coifman and G. Weiss,Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc.83 (1977), 569–645.MATHMathSciNetCrossRefGoogle Scholar
  8. [Fab]
    G. Faber,Über die interpolatorische Darstelung stetige Funktionen, Jahresber. D. M. V.23 (1914), 192–210.Google Scholar
  9. [F-S]
    C. Foias and I. Singer,Some remarks on strongly linearly independent sequences and bases in Banach spaces, Revue Rom. Math. Pures Appl.16(3) (1961), 589–594.MathSciNetGoogle Scholar
  10. [Kor]
    P. P. Korovkin,Linear operators and approximation theory, Fiz.-Mat. Gos. Izdat., Moscow (1959) (in Russian).Google Scholar
  11. [K-P]
    M.I. Kadec and A. Pełczyński,Bases, lacunary sequences and complemented subspaces in the spaces L p, Studia Math.21 (1962), 161–176.MATHMathSciNetGoogle Scholar
  12. [Pri]
    Al. A. Privalov,On the growth of degrees of polynomial basis and approximation of trigonometric projections, Mat. Zametki42(2) (1987), 207–214 (in Russian).MATHMathSciNetGoogle Scholar
  13. [Pri1]
    Al. A. Privalov,On the growth of degrees of polynomial bases, Mat. Zametki48(4) (1990), 69–78 (in Russian).MathSciNetGoogle Scholar
  14. [Sh1]
    B. Shekhtman,On the norms of interpolating operators, Israel J. Math.64(1) (1989), 39–48.CrossRefMathSciNetGoogle Scholar
  15. [Sh2]
    B. Shekhtman,On polynomial “interpolation” in L 1, preprint.Google Scholar
  16. [S-W]
    E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.Google Scholar
  17. [Tor]
    A. Torchinsky,Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.MATHGoogle Scholar
  18. [Ul1]
    P. L. Ulianov,On some solved and unsolved problems in the theory of orthogonal series, Proc. IV All Union Math. Congress2, Publishing House AN SSSR, Moscow (1964), 694–704 (in Russian).Google Scholar
  19. [Wo1]
    P. L. Ulianov,On some results and problems in the theory of bases, Zapiski LOMI, 274–283 (in Russian).Google Scholar
  20. [Wo2]
    P. Wojtaszczyk,The Franklin system is an unconditional basis in H 1, Arkiv für Mathematik20(2) (1982), 293–300.MATHCrossRefMathSciNetGoogle Scholar
  21. [Wo2]
    P. Wojtaszczyk,Banach Spaces for Analysts, Cambridge University Press (to appear).Google Scholar

Copyright information

© Hebrew University 1991

Authors and Affiliations

  • P. Wojtaszczyk
    • 1
  • K. Woźniakowski
    • 2
  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland

Personalised recommendations