Stability ofMultiscale Transformations

349

via a limit process, which is often called subdivision scheme [Dl]. However, in many cases a biorthogonal system with the above properties can be constructed by modifying an initially given system defined on all of M", say. In this case one gets around a possibly difficult convergence analysis and Theorem 3.1 or Theorem 3.3 applies directly. Examples of such circumstances are encountered when establishing stability for wavelet bases adapted to bounded domains [CDD], when constructing compactly supported divergence-free wavelets (see, e.g., [U]), or when constructing pressure and velocity trial spaces with built in Ladysenskaja-Babuska-Brezzi condition for the Stokes lem [DKU].

4 . Proof of Theorem 3.3

Our goal is to show that under the assumptions of Theorem 3.3 one can always construct a subspace (7 of V for which the hypotheses of Theorem 3.2 are satisfied so that the assertion of Theorem 3.3 follows from Theorem 3.2.

4 . 1 . Spaces Induced by Q

As candidates for U in Theorem 3.2 we introduce the following scale of auxiliary spaces. Fix some ^ > 1 and let Aq for r > 0 denote the space of those elements in V for which

m\A^^

Y . o'''Ш - Qj-iH\\ (4.1)

V7=0 /

is finite. Standard arguments confirm that Aq is a Banach space (see, e.g., [BS] or [BL, ma 2.2.1]).

Remark 4.1. The spaces A^ are reflexive (relative to the duality pairing induced by the inner product (♦,•) on У ). D

Proof . Defining

( i ; , w),:=Yl Q^'J {(Qj - Qj-i)v. {Qj - Qj,,)w),

7=0

this can be deduced from the fact that Aq is a Hilbert space relative to (•, •)r. Alternatively, one can argue as follows. Since by [T, Lemma 1.11.1], one has for

hiV ) := {V = {vj]jmo -VjeV. ||v||,,(V) := \f^ \\Vj\\l\ < oo} that {liiV))' = hiV), the spaces iiiV) are reflexive. Next note that the mapping

is an injective bounded linear operator from Aq into iiiy)- Since Aq is a Banach space,

f / , :=a,(A^^)

is a closed subspace of ^2(^) and hence also reflexive. Since g^ is an isometry onto Ux, the assertion follows. D

Remark 4.2. For any r > 0 one has

A^Q ^ V (4.2)