  The class of cartoon-like functions, classicly defined as piecewise $C^2$
functions consisting of smooth regions separated by $C^2$ discontinuity curves,
is a well-established model for image data. The quest for optimal approximation
of this class has among others led to the development of curvelets,
contourlets, and shearlets. Due to parabolic scaling, these systems are able to
provide a quasi-optimal $N$-term approximation rate of order $N^{-2}$.
Replacing parabolic scaling by $\alpha$-scaling, one obtains $\alpha$-curvelets
and $\alpha$-shearlets, which interpolate between wavelet-type systems
($\alpha=1$), parabolically scaled systems ($\alpha=\frac12$), and
ridgelet-type systems ($\alpha=0$). Previous research shows that in the range
$\alpha\in[\frac{1}{2},1)$ they provide quasi-optimal approximation for
cartoons of regularity $C^{1/\alpha}$ with a rate of order $N^{-1/\alpha}$.
  In this work we continue to explore $\alpha$-scaled representation systems,
with the aim to better understand the role of the parameter $\alpha$ for
approximation. Concerning $\alpha$-curvelets with $\alpha<1$, we prove that the
best possible $N$-term approximation rate achievable for cartoons with curved
edges is limited to at most $N^{-1/(1-\alpha)}$, independent of the smoothness
of the cartoons. The maximal rate achievable by simple thresholding of the
frame coefficients is even bounded by $N^{-1/\max\{\alpha,1-\alpha\}}$. If the
edges of the cartoons are straight the approximation performance of
$\alpha$-curvelets is different: Assuming $C^\beta$ regularity, we establish an
approximation rate of order $N^{-\min\{\alpha^{-1},\beta\}}$, which is
quasi-optimal if $\alpha\in [0,\beta^{-1}]$. Finally, via the framework of
$\alpha$-molecules, the obtained results are extended to other $\alpha$-scaled
systems including in particular $\alpha$-shearlets.

Functional Analysis