APPROXIMATION ORDER PROVIDED BY REFINABLE FUNCTION VECTORS

In this paper we consider L(p)-approximation by integer translates of a finite set of functions phi(v) (v = 0, ..., r - 1) which are not nec essarily compactly supported, but have a suitable decay rate. Assuming that the function vector phi = (phi(v))(v=0)(r=1) is refinable, neces sary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by int eger translates of phi(v), then a factorization of the refinement mask of phi can be given. This result is a natural generalization of the r esult for a single function phi, where the refinement mask of phi cont ains the factor ((1 + e(-iu))/2)(m) if approximation order m is achiev ed.