ABSTRACT
It is also possible to scale with certain integer matrices. This case was studied by Cohen and Daubechies [2]. The shift-invariance of Vo can be interpreted as an invariance property with respect to the action of a discrete subgroup, and the scaling process can be viewed as the action of a certain group automor phism. Therefore, on a general locally compact Abelian group we can define the following:
D efin ition 1.1. Let G be a locally compact Abelian group, and suppose that G contains a discrete countable subgroup K such that the quotient group ^/k is compact. Furthermore, let us assume that there exists an automorphism A of G such that A (K ) C K. Then a multiresolution analysis o f L2(G, me) , where m o denotes the Haar measure on G, is a sequence {Vj } je& ° f closed subspaces of L2(G , mo) such that
(1.3)
(1.4)
(1.5)
There exists a function φ and constants C\, c2 E 1R+ such that Vo is the closed linear span o f o k) and
( 1 .6 )
R em ark 1.2 . It is a well-known fact that a Haar measure, i.e., a translationinvariant Borel measure, exists on a locally compact Abelian group. The Haar measure is unique up to multiplication by constants, see [3] for details.
