Finite Frames Related to Sampling in Finite-Dimensional U-Invariant Subspaces

In this chapter the sampling problem in finite-dimensional U-invariant subspaces of a separable Hilbert space \(\mathcal {H}\) is considered. Since the involved samples are frame coefficients in a suitable Euclidean space \(\mathbb {C}^N\) , the problem reduces to obtain dual frames having a U-invariance property. The obtained sampling theory meets a nice application to periodic extensions of finite signals. The case where \(\{U(g)\,:\, g\in \mathbf {G}\}\) is a unitary representation of a non-abelian finite group \(\mathbf {G}\) which is a knit product of groups on a Hilbert space \(\mathcal {H}\) is also considered.