3.3. Frames of wavelets.

We saw in 3.1 that in order to have a numerically stable reconstruction algorithm for f from the ? f, ? _m,n ?, we require that the ? _m,n constitute a frame. In 3.2 we found an algorithm to reconstruct f from the ? f, ? _m,n ? if the ? _m,n do constitute a frame; for this algorithm the ratio of the frame bounds is important, and we will come back to ways of computing at least a bound on this ratio, later in this section. First, however, we show that the requirement that the ? _m,n constitute a frame already imposes that ? is admissible.

3.3.1. A necessary condition: Admissibility of the mother wavelet.

Theorem 3.3.1.

If the ? _m,n( x) = a ₀ ^{? m/2} ?( a ₀ ^m x ? nb ₀), m, , constitute a frame for with frame bounds A, B, then

and

Proof.

We have, for all ,

If we write (3.3.3) for f = u _?, and add all the resulting inequalities, weighted with coefficients c _? ? 0 such that , then we obtain

In particular, if C is any positive trace-class operator (see Preliminaries), then

where the u _? are orthonormal, c _? ? 0, and . For any such operator, we have therefore, by (3.3.4),
We now apply (3.3.5) to a...