The problem to be examined is that of determining the best n-point formula for integrating a smooth function over a region R of the (x,y)-plane,
$$\iint {_Ru\left( {x,y} \right)}\;dx\;dy \cong \sum\nolimits_{j = 1}^n {{\alpha _j}u\left( {{x_j},{y_j}} \right)} ,$$
where both the weights {a.} and the nodes {(xj,yj.)} are at our disposal. It is assumed that the only a priori information we have concerning u is that the integral over R of a certain quadratic form in its second derivatives is bounded; this is what we mean by calling the function “smooth”. Our work to some extent parallels that of Schoenberg [1] on optimal quadrature.


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  1. [1]
    I. J. Shoenberg, Monosplines and Quadrature Formulae, in Theory and Applications of Spline Functions, ed. T.N.E. Greville (New York, 1969).Google Scholar
  2. [2]
    J. Meinguet, Multivariate Interpolation at Arbitrary Points Made Simple, ZAMP, to appear.Google Scholar
  3. [3]
    J. Duchon, Interpolation des Fonctions de Deux Variables Suivant le Principe de la Flexion des Plaques Minces, RAIRO Analyse Numérique 10 (1976), 5–12.Google Scholar
  4. [4]
    S. Bergman, M. Schiffer, Kernel Functions and Elliptic Partial Differential Equations in Mathematical Physics, (New York, 1953), p.239.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • D. C. Handscomb
    • 1
  1. 1.Oxford University Computing LaboratoryOxfordEngland

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