ABSTRACT
In order to solve the refinement equation (1.1), we start with an initial function (po given by
<j>o(xi, • • • , x s) ■= (®1» • ■ * >*«) 3= 1
where x is the univariate hat function defined by
x(x) := max{l — |a;|,0},
Then we employ the iteration scheme Q%<po, n = 0,1,2, • • • , where Qa is the bounded linear operator on Lp(Rs) (1 < p < oo) given by
Qaf := £ «(/?)/(2-- /? ) , / € LP(RS). (2.5) /3ezs
This iteration scheme is called a subdivision scheme or a cascade algorithm associated with the mask a (see [8, 23]). For any p such that 1 < p < oo, we say that the subdivision scheme associated with a mask a converges in the Lp norm if there exists a function / in Lp(Rs) such that lim^-^oo \\Q2</>o — f\\p = 0. If this is the case, then the limit function / must be the normalized solution of the refinement equation (1.1) with the refinement mask a. It was demonstrated in Theorem 3.4 in [35] that if the shifts of a refinable function 4>a in Lp(Rs) are stable, then the subdivision scheme associated with mask a necessarily converges in the Lp norm.
