Affine Synthesis onto Lp when 0 < p ≤ 1

The affine synthesis operator Sc = ∑j>0 ∑k∈Zd cj,kψj,k is shown to map the coefficient space (Z+ × Zd ) surjectively onto Lp(Rd ), for p ∈ (0, 1]. Here ψj,k(x) = | det aj |ψ(aj x − k) for dilation matrices aj that expand, and the synthesizer ψ ∈ Lp(Rd ) need satisfy only mild restrictions, for example, ψ ∈ L1(Rd ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of Lp follows immediately: ‖f ‖p ≈ inf  ∑ j>0 ∑ k∈Zd |cj,k |p 1/p : f = ∑ j>0 ∑ k∈Zd cj,kψj,k  . Tools include an analysis operator that is nonlinear on Lp .