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 ∈ ℤd, 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.
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Goodman, T.N.T., Lee, S.L. & Tang, W.S. Wavelet bases for a set of commuting unitary operators. Adv Comput Math 1, 109–126 (1993). https://doi.org/10.1007/BF02070823
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DOI: https://doi.org/10.1007/BF02070823
Keywords
- Hilbert space
- commuting unitary operators
- Riesz basis
- wandering subspaces
- multiresolution approximation
- duality principle
- box splines