Multiresolution Analysis On Holomorphic Wavelets

In this article,we mainly discuss the wavelet transform associated to square integrable group representation on L~2(C).Let SL(2,C)= NASU(2)be the I-wasawa decomposition of group SL(2,C)where SU(2)is the maximal compact subgroup of SL(2,C),then P = NA = SL(2,C)/SU(2)can be identified with the quaternionic upper half-plane H_+~c.P is the automorphism group of H_+~c and has a reducible unitary representation on L~2(C).We decompose L~2(C)into the direct sum of the irreducible invariant closed subspace under the unitary represen-tation,and the restrictions of the unitary representation on these subspaces are square integrable.Then the author gave the wavelet transforms on the subspaces where the wavelet transforms leads to isometric operators from the Hardy spaces to L~2(H_+~c,dzdt/t~3).By using the projection operator from L~2(C)into Hardy spaces,we obtain the multiresolution analysis of Hardy spaces via the multiresolution analysis of L~2(C).