Spanning and sampling in Lebesgue and Sobolev spaces

Abstract

We establish conditions under which the small-scale affine system {𝜓(𝑎j 𝑥 - 𝑘) : j ≥ J, 𝑘 ∈ℤᵈ}, with J∈ℤ fixed, spans the Lebesgue space 𝐿ᵖ(ℝᵈ) and the Sobolev space 𝑊ᵐ,ᵖ(ℝᵈ), 1 ≤ 𝑝 < ∞. The dilation matrices 𝑎j are expanding (meaning limj →∞ ll𝑎j ⁻¹ ll = 0) but they need not be diagonal. For spanning 𝐿ᵖ we require ∫ℝᵈ 𝜓𝑑𝑥 ≠ 0 and (when 𝑝 > 1) that the periodization of l𝜓l or of 𝟙{𝜓≠₀} be bounded. To span 𝑊ᵐ,ᵖ we also require a Strang-Fix condition on 𝜓. But we impose this condition only to order 𝑚 - 1, whereas earlier authors required order 𝑚. Our spanning results are derived from explicit "Shannon" type sampling formulas that express an arbitrary function 𝑓 as a limit of linear combinations of the 𝜓(𝑎j 𝑥 - 𝑘). The coefficients in these sampling formulas are local averages of 𝑓, or pointwise values off when 𝑓 has some regularity.