Approximation Order Provided by Reenable Function Vectors

In this paper we considerLp approximation by integer trans lates of a nite set of functions r which are not nec essarily compactly supported but have a suitable decay rate Assuming that the function vector r is re nable necessary and su cient conditions for the re nement mask are derived In particular if algebraic polynomials can be exactly reproduced by integer translates of then a factorization of the re nement mask of can be given This result is a natural generalization of the result for a single function where the re nementmask of contains the factor e iu m if approximation order m is achieved Introduction Recently a lot of papers have studied the so called multiresolution analysis of multiplicity r r IN r generated by dilates and translates of a nite set of functions r and the construction of corresponding mul tiwavelets cf e g Donovan Geronimo Hardin and Massopust Goodman Lee and Tang Herv e Plonka In Alpert multiwavelets are used for sparse representation of integral op erators Further applications for solving di erential equations by nite element methods seem to be possible since scaling functions and multiwavelets with very small support can be constructed cf e g Plonka For nite elements short support is crucial In order to obtain multiwavelets with vanishing mo ments the problem remains how to choose the scaling functions such that algebraic polynomials of degree m m IN can be exactly reproduced by a linear combination of integer translates of r or such Date received Date revised Communicated by AMS classi cation A A A C