ABSTRACT
We are confronted with the task of producing scaling functions φ: R —► C having the following properties: (a)
(b)
If all these conditions are met, then Theorem (5.13) will provide us with an orthonormal basis of wavelets having compact support. Condition (a) immediately implies φ E L 1 and φ E C°°; furthermore, we know from (5.6) that only finitely many hk are nonzero. It follows that the generating function
is a trigonometric polynomial satisfying the identity
(1) and having the special values H( 0 ) = 1 , Η(π) = 0 ; see (5.10). The systematic construction of polynomials with these properties is an alge braic problem that we shall take up in the next section. For the moment we assume that we have such an H at our disposal, and we begin our undertaking by showing that the corresponding scaling function φ, if there is one at all, is uniquely determined by H. Applying (b) recursively r times we obtain
and, therefore, because of (c),
(2)
if the infinite product converges. In this regard, we show:
(6 .1 ) Assume that the generating function H € C 1 satisGes the identity (1) as well as H(0) = 1. Then the product (2) converges locally uniformly on R to a function φ e L2.
