  It is well-known that curvelets provide optimal approximations for so-called
cartoon images which are defined as piecewise $C^2$-functions, separated by a
$C^2$ singularity curve. In this paper, we consider the more general case of
piecewise $C^\beta$-functions, separated by a $C^\beta$ singularity curve for
$\beta \in (1,2]$. We first prove a benchmark result for the possibly
achievable best $N$-term approximation rate for this more general signal model.
Then we introduce what we call $\alpha$-curvelets, which are systems that
interpolate between wavelet systems on the one hand ($\alpha = 1$) and curvelet
systems on the other hand ($\alpha = \frac12$). Our main result states that
those frames achieve this optimal rate for $\alpha = \frac{1}{\beta}$, up to
$\log$-factors.

Functional Analysis