Wavelet coefficients, quadrature mirror filters and the group SO(4)

Let c=(c/sub j/)/sub j/spl isin//sub //Z/spl isin/l/sup 2/ and u(/spl xi/) be its Fourier series. We consider the local or gauge group of linear transformations preserving the combined energy spectral density at angular frequencies offset by /spl pi/, |u(/spl xi/)|/sup 2/+|u(/spl xi/+/spl pi/)|/sup 2/, separately at each /spl xi//spl isin/[0,/spl pi/]. This gauge transformation group consists of the mappings from [0,/spl pi/]/spl rarr/SO(4), the group of 4-dimensional rotations. It is shown that the usual decomposition of a discrete sequence into scale and wavelet coefficients is a consequence of the local decomposition of the 4-dimensional (vector) represention of the group SO(4) as a tensor product of the half-spin representations of its associated spin group, Spin(4).