Selçuk Journal
of
Applied Mathematics
WinterSpring, 2001
Volume 2
Number 1
Research Center
of
Applied Mathematics

SJAM
WinterSpring 2001, Volume 2  Number 1

Hierarchical Cubature Formulas

Vladimir Vaskevich 
Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia
email : vask@math.nsc.ru
Received: March 10, 2001

Summary
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 wellknown FaberSchauder 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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