2 1. MAIN RESULTS An axiomatic development of frame decompositions of the identity operator in the context of integrable, irreducible, unitary representations of a locally compact group has been developed in [FG1], [FG2] and [Gro]. By specializing to the affine group, one obtains frame decompositions of classical Banach spaces on R n by identifying these spaces as co-orbit spaces associated with Banach lattices on the upper half-plane R+ ' . Smoothness of both the analyzing and synthesizing functions in the decomposition, a key feature of this memoir, however, cannot be guaranteed with this axiomatic approach. The theory of molecules and their applications to singular integrals developed in this memoir began with two separate efforts, one centered at the University of Texas at Austin, the other at Washington University. Initially, each had a different emphasis, but for the most part there was a considerable overlapping of ideas. Once the extent of the overlap was realized, it was decided that one joint publication would best represent the efforts of the two groups. 1. Frame decompositions . A countable family of vectors {^A}AGA m a s e P a r a b l e Hilbert space H is said to be a frame when there exist positive constants C\ and C2 such that (i-i) c.Wgf Y,\(9^x)\2 c2\\g\\2 xeA for all g £ Ti. Because of Parseval's identity a frame is a natural generalization of the notion of orthonormal basis. The constants C\ and C2 in (1.1) are called the frame bounds and the condition itself is a stability condition because the coefficients (g,(p\) depend continuously on g. Any f in H can be reconstructed from the ip\ using the frame operator A G A associated with the {^AJAGA ([Dau, HeW]). Indeed, (1.1) ensures that C is bounded and invertible on H with operator norms WC-'w l/d , ||£|| c 2 controlled by the frame bounds in fact, C is bounded and invertible on H if and only if (1.1) holds. Thus the vectors p\ = £~1(p\ again belong to H and form a frame {/A}AGA called the dual frame to {^A}AGA- The reconstruction of / in terms of the (f\ is then given by / = ^(f,P\)P\ = ^(f,P\)p\ AEA AGA In particular, the set {(^A}AGA 1S c o m p l e t e in H in the sense that finite linear combinations of the ip\ form a dense subspace of Ti. So, just as with orthonormal bases, frames give rise to stable reproducing formulas. To extend the definition of a frame to a Banach space B the equivalent version of (1.1) is not used because the f2-sequence norm on the coefficients does not always have an obvious replacement for a general B whereas invertibility of an operator always makes sense. Nevertheless, if B is assumed to be, say, a co-orbit space
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