ON THE SQUARE-INTEGRABLE IRREDUCIBLE REPRESENTATIONS ON SEMI-DIRECT PRODUCT GROUPS

Higher dimensional analogs of the classical continuous wavelet transform are developed for Euclidean spaces whose dimension is a perfect square.The space of all n n real matrices can be identified with Rn2 as an additive abelian group.The group of invertible n n real matrices naturally acts on this abelian group by matrix multiplication.The resulting semidirect product group forms a group of affine transformations of Rn2 which may be viewed as a generalization of the affine transformations of Rn2 whose uniquesquare-integrable representation underlies the classical one-dimensional continuous wavelet transform. A continuous wavelet transform for Rn2 is derived and its specific details are worked out for R4 resulting in a 4D continuous wavelet transform.We provide three equivalent versions of the irreducible representation of G=A⋊H for A=M(n,R) and H=GL(n,R) underlies the continuous wavelet transform and then three versions of the irreducible representation of G=A⋊H such that A is locally compact abelian group and H is locally compact group.

کلیدواژه ها:

representation ، semi-direct product ، wavelet transform