h1

% IMPORTANT: The following is UTF-8 encoded.  This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.

@PHDTHESIS{Burtscheidt:808437,
      author       = {Burtscheidt, Achim Thomas},
      othercontributors = {Führ, Hartmut and Rauhut, Holger},
      title        = {{V}anishing moments conditions for atomic decompositions of
                      coorbit spaces on quasi-{B}anach spaces},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      reportid     = {RWTH-2020-12061},
      pages        = {1 Online-Ressource (xiv, 162 Seiten) : Illustrationen,
                      Diagramme},
      year         = {2020},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University 2021; 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.},
      cin          = {114510 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114510_20190627$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2020-12061},
      url          = {https://publications.rwth-aachen.de/record/808437},
}