The homogeneous approximation property for wavelet frames (Q2643851)

For a sequence \(\Lambda\) of points in \(\mathbb{R}^+\times \mathbb{R}\) and a function \(\psi\in L^2(\mathbb{R})\), an irregular wavelet frame is defined to be \(W(\psi, \Lambda)=\{ a^{-1/2}\psi(\frac{x}{a}-b)\}_{(a,b)\in \Lambda}\). Theorem 4.2, the main result of this paper, states that if \(\psi \in B_0\) (see Definition 3.5) and if \(W(\psi,\Lambda)\) is a frame for \(L^2(R)\), then \(W(\psi, \Lambda)\) satisfies the strong Homogeneous Approximation Property (HAP) (see Definition 4.1). It is also shown in Theorem 4.3 that the HAP has several interesting implications on the geometry of \(\Lambda\). In particular a relationship is derived between the affine Beurling density of an irregular wavelet frame and the affine Beurling density of any other irregular Riesz wavelets.