ABSTRACT
We begin wi th a scaling function 0, a real valued function on R which is r times differentiate and whose derivatives are continuous and rapidly decreasing. That is, cj) satisfies
\^k)(t)\ < Cpk(l + \t\)~ p, k = 0 , 1 , . . . , r, p G Z, t G R, (3.1)
and hence (j) £ Sr. In order for it to qualify as a scaling function, there must be associated wi th cj) a multiresolution analysis of L 2 ( R ) , i.e., a nested sequence of closed subspaces {Vmjm^z such that
r (i) {(pit — n)} is an orthonormal basis of Vo
(ii) • • • c y _ i C V b c y i C - - - c L 2 ( R )
(3.2) (Hi) f G Vm <^ /(2-) G Vm+1
nmVm = {0h {JmVm=L*(R).
