Abstract
An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.
Highlights
Statement of the ProblemAn abstract sampling theory associated with non abelian groups is derived for the specific case of a unitary representation of a semi-direct product of groups on a separable Hilbert space
An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied
The case where G is a discrete locally compact abelian (LCA) group and the samples are taken at a uniform lattice of G has been solved in Ref. [10]; this work relies on the use of the Fourier analysis in the LCA group G
Summary
An abstract sampling theory associated with non abelian groups is derived for the specific case of a unitary representation of a semi-direct product of groups on a separable Hilbert space. Symmetry 2019, 11, 529 where 1 H denotes the identity element in H These samples are nothing but a generalization of average sampling in shift-invariant subspaces of L2 (Rd ); see, among others, Refs. Roughly speaking, substituting the output of the synthesis part in x = ∑(n,h)∈G α(n, h) U (n, h) a, we will obtain the corresponding sampling formula in A a This said, as it could be expected, the problem can be mathematically formulated as the search of dual frames for ( G ) having the form. The paper ends with an illustrative example involving the quasi regular representation of a crystallographic group on L2 (Rd ); sampling formulas involving average or pointwise samples are obtained for the corresponding U-invariant subspaces in L2 (Rd )
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