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

Let c=(c_j)_{j∈ Z}∈l² and u(ξ) 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 π, |u(ξ)|²+|u(ξ+π)|², separately at each ξ∈[0,π]. This gauge transformation group consists of the mappings from [0,π]→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)