Wavelet bases for a set of commuting unitary operators

AbstractLet (U=U1, ...,Ud) be an orderedd-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space ℋ, and letX:={x1, ...,xr} ⊂ ℋ such that $$U^{\mathbb{Z}^d } X: = \{ U_1^{n_1 } \ldots U_d^{n_d } x_j :(n_1 , \ldots ,n_d ) \in \mathbb{Z}^d ,j = 1, \ldots ,r\} $$ is a Riesz basis of the closed linear spanV0 of $$U^{\mathbb{Z}^d } X$$ . Suppose there is unitary operatorD on ℋ such thatV0 ⊂DV0 =:V1 andUnD=DUAn for alln ∈ ℤd, whereA is ad ×d matrix with integer entries and Δ := det(A) ≠ 0. Then there is a subset Λ inV1, withr(Δ − 1) vectors, such that $$U^{\mathbb{Z}^d } (\Gamma )$$ is a Riesz basis ofW0, the orthogonal complement ofV0 inV1. The resulting multiscale and decomposition relations can be expressed in a Fourier representation by one single equation, in terms of which the duality principle follows easily. These results are a consequence of an extension, to a set of commuting unitary operators, of Robertson's Theorems on wandering subspace for a single unitary operator [24]. Conditions are given in order that $$U^{\mathbb{Z}^d } (\Gamma )$$ is a Riesz basis ofW0. They are used in the construction of a class of linear spline wavelets on a four-direction mesh.