##
**Optimally sparse multidimensional representation using shearlets.**
*(English)*
Zbl 1197.42017

Summary: In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions \(f\) which are \(C^2\) except for discontinuities along \(C^2\) curves. More specifically, if \(f_N^S\) is the \(N\)-term reconstruction of \(f\) obtained by using the \(N\) largest coefficients in the shearlet representation, then the asymptotic approximation error decays as
\[
\| f-f_N^S\|_2^2 \asymp N^{-2} (\log N)^3,\quad N \to \infty,
\]
which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate \(N^{-1}\) associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.

### MSC:

42C15 | General harmonic expansions, frames |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |