Advertisement

The Lp minimality and near-minimality of orthogonal polynomial approximation and integration methods

  • J. C. Mason
Contributors
Part of the Lecture Notes in Mathematics book series (LNM, volume 1329)

Abstract

It is known that the Chebyshev polynomials of the first and second kinds are minimal in Lp on [−1,1] with respect to appropriate weight functions, namely certain powers of 1−x2, for 1≤p≤∞. These properties are here exploited in two applications. First, convergence and optimality properties are established for a "complete" Chebyshev polynomial expansion method for the determination of indefinite integrals. Second, conjectures are derived concerning the near-minimality of the Laguerre polynomials L n ±1/2 (2β ×) for β≃1 with respect to appropriate exponentially weighted Lp norms on [0,∞).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.C. MASON, Some properties and applications of Chebyshev polynomial and rational approximation. In: "Rational Approximation and Interpolation". P. Graves-Morris, E.B. Saff, and R.S. Varga (Eds.), Springer-Verlag, Berlin, 1984, pp. 27–48.CrossRefGoogle Scholar
  2. 2.
    J.C. MASON, Near-minimax approximation and telescoping procedures based on Laguerre and Hermite polynomials. In: "Polynômes Orthogonaux et Applications", C. Brezinski, A. Draux, A.P. Magnus, P. Maroni et A. Ronveaux (Eds.). Springer-Verlag, Berlin, 1985, pp. 419–425.CrossRefGoogle Scholar
  3. 3.
    S. FILIPPI, Angenäherte Tschebyscheff-Approximation einer Stammfunktion-eine Modifikation des Verfahrens von Clenshaw und Curtis. Numer. Mathematik 6 (1964), 320–328.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I. SLOAN and W.E. SMITH, Properties to interpolating product integration rules. SIAM J. Numer. Anal. 19 (1982), 427–442.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C.W. CLENSHAW and A.R. CURMIS, A method for numerical integration on an authomatic computer. Numer. Mathematik 2 (1960), 197–205.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • J. C. Mason
    • 1
  1. 1.Computational Mathematics GroupRoyal Military College of ScienceShrivenham, SwindonEngland

Personalised recommendations