Affine synthesis onto L ^p when 0 < p≤ 1

The affine synthesis operator Sc=∑_j>0, ∑_kℤ^d c_j,kε_j,k is shown to map the coefficient space ℓ^p (ℤ₊×ℤ ^d ) surjectively onto L ^p (ℝ ^d ), for p (0,1]. Here ψ _j,k (x)=|det∈a _j |^1/p ψ(a _j x-k) for dilation matrices a _j that expand, and the synthesizer ψ L ^p (ℝ ^d ) need satisfy only mild restrictions, for example, ψ L ¹(ℝ ^d ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L ^p follows immediately: ∥∫∥ ∑ _j>0,∑_{kℤc_j,k}|^p{1/p}:f= ∑_j>0,∑_k ℤ ^dc _j,kψ_j,k∼ Tools include an analysis operator that is nonlinear on L ^p .