Selšuk Journal of Applied Mathematics

 Selšuk Journal of
  Applied Mathematics

  Winter-Spring, 2001
  Volume  2
  Number 1

Research Center of 
  Applied Mathematics

 SJAM Winter-Spring 2001, Volume 2 - Number 1

Hierarchical Cubature Formulas

Vladimir Vaskevich

Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia
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Received: March 10, 2001

We study the properties of hierarchical bases in the space of continuous functions with bounded domain and construct the hierarchical cubature formulas. Hierarchical systems of functions are similar to the well-known Faber-Schauder basis. It is shown that arbitrary hierarchical basis generates a scale of Hilbert subspaces in the space of continuous functions. The scale in many respects is similar to the usual classification of functional spaces with respect to moothness. By integration over initial domain the standard interpolation formula for the given continuous integrand, we construct the hierarchical cubature formulas and prove that each of these formulas is optimal simultaneously in all Hilbert subspaces associated with the initial hierarchical basis. Hence, we have constructed the universally optimal cubature formulas.

Key words
calculation of integrals, guaranteed accuracy, cubature formulas, bases in Sobolev spaces, hierarchical bases

Mathematics Subject Classification (1991): 65D32, 65D30, 41A55

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