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

Burtscheidt, Achim Thomas; Führ, Hartmut (Thesis advisor); Rauhut, Holger (Thesis advisor)

Aachen (2020, 2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

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 $p<1$). However, since it is desirable to have compactly supported analyzing vectors, and since we can directly construct compactly supported vectors with vanishing moments, we develop criteria using this access. So far, vanishing moments results for quasi-Banach spaces are just known for some special cases. We will expand known vanishing moments results for Banach spaces ($p\geq1$) to quasi-Banach spaces in a very general way. We will derive sufficient criteria for coorbit spaces, which admit control weights of the form $v(x,y)=(1+|x|)^kg(h)$. Moreover, we study the asymptotic behavior of vanishing moments. We derive sufficient control weights for any $L^p(G)$ with $p>0$ 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.