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Constructive Approximation

, Volume 3, Issue 1, pp 363–375 | Cite as

Periodic monosplines and perfect splines of least norm

  • B. D. Bojanov
  • Daren Huang
Article

Abstract

LetP(N,m;r1,...,r n ) be the class of 1-periodic perfect splines of degreem with 2N knots, which haven distinct zeros in one period with multiplicitiesr1,...,r n , respectively. We show that there exists a unique extremal elementP*P(N,m;r1,...,r n ) of minimal uniform norm which equioscillates. This problem is related to the optimal recovery of smooth periodic functions on the basis of the Hermitian data.

Key words and phrases

Splines of least norm Perfect splines Mono splines Multiple zeros 

AMS classification

41A15 41A29 41A52 

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References

  1. 1.
    R. B. Barrar, H. L. Loeb (1978):On monosplines with odd multiplicity of least norm. J. Analyse Math.,33:12–38.Google Scholar
  2. 2.
    R. B. Barrar, H. L. Loeb (1984):Optimal monosplines with maximal number of zeros. SIAM J. Math. Anal.,15.Google Scholar
  3. 3.
    B. D. Bojanov (1977):A note on the optimal approximation of smooth periodic functions. C. R. Acad. Bulgare Sci.,30:809–812.Google Scholar
  4. 4.
    B. D. Bojanov (1980):Perfect splines of least uniform deviation. Anal. Math.,6:185–197.Google Scholar
  5. 5.
    D. Braess, N. Dyn (1982):On the uniqueness of monosplines and perfect splines of least L 1-and L 2-norm. J. Analyse Math.,41:217–233.Google Scholar
  6. 6.
    C. H. Fitzgerald, L. L. Schumaker (1969):A differential equation approach to interpolation at extremal points. J. Analyse Math.,22:117–134.Google Scholar
  7. 7.
    C. R. Hobby, J. R. Rice (1965):A moment problem in L 1 approximation. Proc. Amer. Math. Soc.,16:665–670.Google Scholar
  8. 8.
    D. Huang (to appear):Perfect spline with least L 1 norm.Google Scholar
  9. 9.
    C. A. Micchlli (1977):Best L 1-approximation by weak Chebyshev systems and the uniqueness of interpolating perfect splines. J. Approx. Theory,19:1–14.Google Scholar
  10. 10.
    L. L. Schumaker (1976):Zeros of spline functions and applications. J. Approx. Theory.18:152–168.Google Scholar
  11. 11.
    A. A. Žensykbaev (1977):Best quadrature formula for some classes of periodic differentiable functions. Math. USSR-Izv.,11:1055–1071. (Russian original Izv. Akad. Nauk SSSR Ser. Mat.,41 (1977), 1110–1124.)Google Scholar
  12. 12.
    A. A. Žensykbaev (1981):Monosplines of minimal norm and the best quadrature formulae. Russian Math. Surveys,36(4):121–180. (Russian original: Uspekhi Mat. Nauk,36(4) (1981), 107–159.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • B. D. Bojanov
    • 1
  • Daren Huang
    • 2
  1. 1.Department of MathematicsUniversity of SofiaSofiaBulgaria
  2. 2.Department of MathematicsZhejiang UniversityHangzhouPeople's Republic of China

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