Quadrature in Besov spaces on the Euclidean sphere

  • Authors:

  • K. Hesse;H. N. Mhaskar;I. H. Sloan

  • Affiliations:

  • School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia;Department of Mathematics, California State University, Los Angeles, California 90032, USA;School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

  • Venue:
  • Journal of Complexity
  • Year:
  • 2007

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Abstract

Let q=1 be an integer, S^q denote the unit sphere embedded in the Euclidean space R^q^+^1, and @m"q be its Lebesgue surface measure. We establish upper and lower bounds forsupf@?B"p","@r^@c@!"S"^"qfd@m"q-@?k=1Mw"kf(x"k),x"k@?S^q,w"k@?R,k=1,...,M,where B"p","@r^@c is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of x"k and w"k that admit exact quadrature for spherical polynomials of a given degree, and satisfy a certain continuity condition; the lower bounds are obtained for the infimum of the above quantity over all choices of x"k and w"k. Since the upper and lower bounds agree with respect to order, the complexity of quadrature in Besov spaces on the sphere is thereby established.