Density of the Set of Generators of Wavelet Systems

Given a function $\psi$ in ${\text{LL}}^2({\mathbb{R}}^d)$, the affine (wavelet) system generated by $\psi$, associated to an invertible matrix $a$ and a lattice $\text{zG}$, is the collection of functions ${|\det a|^{j/2} \psi(a^jx-\gamma): j \in \Z, \gamma \in \zG}$. In this article we prove that the set of functions generating affine systems that are a Riesz basis of ${\text{LL}}^2({\mathbb{R}}^d)$ is dense in ${\text{LL}}^2({\mathbb{R}}^d)$. We also prove that a stronger result is true for affine systems that are a frame of ${\text{LL}}^2({\mathbb{R}}^d)$. In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of ${\text{LL}}^2({\mathbb{R}}^d)$ with the induced metric. As a byproduct we introduce the $p$-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems.