IUMJ

Title: Affine synthesis onto Lebesgue and Hardy spaces

Authors: H.-Q. Bui and R. S. Laugesen

Issue: Volume 57 (2008), Issue 5, 2203-2234

Abstract:

The affine synthesis operator $Sc = \sum_{j > 0} \sum_{k \in \mathbb{Z}^{d}} c_{j,k} \psi_{j,k}$ is shown to map the mixed-norm sequence space $\ell^{1}( \ell^{p} )$ surjectively onto $L^{p}( \mathbb{R}^{d} )$ under mild conditions on the synthesizer $\psi \in L^{p}( \mathbb{R}^{d} )$, $1 \leq p < \infty$, with $\int_{\mathbb{R}^{d}} \psi \,\mathrm{d}x = 1$. Here \[ \psi_{j,k}(x) = | \det a_{j} |^{1/p} \psi( a_{j}x - k ), \] and the dilation matrices $a_{j}$ expand, for example $a_{j} = 2^{j}I$. Affine synthesis further maps a discrete mixed Hardy space $\ell^{1}( h^{1} )$ onto $H^{1}( \mathbb{R}^{d} )$. Therefore the $H^{1}$-norm of a function is equivalent to the infimum of the norms of the sequences representing the function in the affine system: \[ \|f\|_{H^{1}} \approx \inf \Big\{ \sum_{j > 0} \sum_{k \in \mathbb{Z}^{d}} (|c_{j,k}| + |(c*z)_{j,k}|) : f = \sum_{j>0} \sum_{k \in \mathbb{Z}^{d}} c_{j,k} \psi_{j,k} \Big\} \] where $z = \{z_{\ell} \}$ is a discrete Riesz kernel sequence.