Wavelet bases for a set of commuting unitary operators

Abstract

Let (U=U ₁, ...,U _d ) be an orderedd-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space ℋ, and letX:={x ₁, ...,x _r } ⊂ 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 spanV ₀ of$U^{\mathbb{Z}^d } X$. Suppose there is unitary operatorD on ℋ such thatV ₀ ⊂D V ₀ =:V ₁ andU ⁿ D=DU ^An for alln ∈ ℤ^d, whereA is ad ×d matrix with integer entries and Δ := det(A) ≠ 0. Then there is a subset Λ inV ₁, withr(Δ − 1) vectors, such that$U^{\mathbb{Z}^d } (\Gamma )$ is a Riesz basis ofW ₀, the orthogonal complement ofV ₀ inV ₁. 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 ofW ₀. They are used in the construction of a class of linear spline wavelets on a four-direction mesh.