Vanishing moments conditions for atomic decompositions of coorbit spaces on quasi-Banach spaces

In this thesis, we derive sufficient and necessary criteria for analyzing vectors in the class of wavelet coorbit spaces $\Co(L^p(\Rd\rtimes H))$ using the notion of vanishing moments. More precisely, we consider wavelet coorbit spaces associated to a square-integrable, irreducible quasi-regular representation of the semi-direct product $G=\Rd\rtimes H$ on $L^2(\Rd)$. The group $G$ consists of affine mappings with dilation taken from an admissible dilation group $H$, which admits an invertible wavelet transform. Under certain conditions, analyzing vectors induce an atomic decomposition of their coorbit space. It is already known that there is a class of non-compactly supported bandlimited Schwartz functions, which are analyzing vectors for all of such wavelet coorbit spaces (even for $p0$ as well as a lower bound for such control weights. It turns out that $\sim\frac1p$ vanishing moments are sufficient for analyzing vectors. Moreover, we see that analyzing vectors (and even good vectors) together with some mild regularity assumptions necessary have vanishing moments of order $\sim\frac1p$. This implies that one cannot find a compactly supported universal wavelet, which is analyzing for all $p>0$ simultaneously