Affine synthesis onto lebesgue and hardy spaces

The affine synthesis operator Sc = Σ_j>0 Σ_k∈ℤd C _j,k Ψ _j,k is shown to map the mixed-norm sequence space ℓ¹(ℓ^p) surjectively onto Lp(Kd) under mild conditions on the synthesizer ψ ∈ L^p(ℝ^d), 1 ≤ p < ∞, with ∫_ℝd ψ dx = 1. Here ψ j,k(x) = |det a j| 1/p ψ(ajx - k), and the dilation matrices ft/ expand, for example aj = 2 ^jI. Affine synthesis further maps a discrete mixed Hardy space ℓ¹ (h¹) onto H¹(ℝ^d). Therefore the H¹-norm of a function is equivalent to the infimum of the norms of the sequences representing the function in the affine system: (Equation Presented) where z = {zℓ} is a discrete Riesz kernel sequence.