Results in Mathematics

, Volume 11, Issue 1–2, pp 144–164 | Cite as

Interpolation on the sphere and bounds for the Lagrangian square sums

  • M. Reimer


For spaces of polynomial functions on the sphere which are invariant against rotation, the square-sums of the Lagrangians can be estimated by means of the smallest eigenvalue of a positive definite system matrix defined by the reproducing kernel and the nodes used. As a consequence, bounds for the corresponding Lebesgueconstants are obtained. There are examples where the method leads to an estimation of the square-sum by one, which cannot be improved. In this case the Lagrangians perform an extremal basis.


Fundamental System Interpolation Operator Nodal System Lebesgue Constant Regular Polyhedron 
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  1. [1]
    G. G. Lorentz: Approximation of functions. Holt, Rinehart and Winston: New York etc. 1966Google Scholar
  2. [2]
    C. Müller: Spherical harmonics. Springer: Berlin, Heidelberg, New York 1966MATHGoogle Scholar
  3. [3]
    D. L. Ragozin: Uniform convergence of spherical harmonic expansions. Math. Ann. 195 (1972) 87–94MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    M. Reimer: Extremal bases for normed vector spaces. In E. W. Cheney (ed.), Approximation Theory, Academic Press: New York 1980, 723–728Google Scholar
  5. [5]
    D. Siepmann, B. Sündermann: On a minimal property of cubic periodic Lagrangian splines. J. Approx. Th. 39 (1983) 236–240MATHCrossRefGoogle Scholar
  6. [6]
    B. Sündermann: On a minimal property of trigonometric interpolation at equidistant nodes. Computing 27 (1981) 371–372MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    B. Sündermann: Projektionen auf Polynomräume in mehreren Veränderlichen. Diss. Dortmund 1983Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1987

Authors and Affiliations

  • M. Reimer
    • 1
  1. 1.Lehrstuhl Mathematik III Fachbereich MathematikUniversität DortmundDortmund 50

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