The structure of shift-invariant subspaces of Sobolev spaces

We analyze shift-invariant spaces V_s , subspaces of Sobolev spaces H^s(ℝ^n) , s∈ℝ , generated by a set of generators φ_i , i∈ I , with I at most countable, by the use of range functions and characterize Bessel sequences, frames, and the Riesz basis of such spaces. We also describe V_s in terms of Gramians and their direct sum decompositions. We show that f∈𝒟_L^2'(ℝ^n) belongs to V_s if and only if its Fourier transform has the form f̂=∑_i∈ If_ig_i , f_i=φ̂_i∈ L_s^2(ℝ^n) , {φ_i( ·+k) k∈ℤ^n, i∈ I} is a frame, and g_i=∑_k∈ℤ^na_k^ie^-2π√(-1) ⟨ · ,k⟩ , with (a^i_k)_k∈ℤ^n∈ℓ^2(ℤ^n) . Moreover, connecting two different approaches to shift-invariant spaces V_s and 𝒱^2_s , s>0 , under the assumption that a finite number of generators belongs to H^s∩ L^2_s , we give the characterization of elements in V_s through the expansions with coefficients in ℓ_s^2(ℤ^n) . The corresponding assertion holds for the intersections of such spaces and their duals in the case where the generators are elements of 𝒮(ℝ^n) . We then show that ⋂_s>0V_s is the space consisting of functions whose Fourier transforms equal products of functions in 𝒮(ℝ^n) and periodic smooth functions. The appropriate assertion is obtained for ⋃_s>0V_-s .