Ridgelets and the Representation of Mutilated Sobolev Functions

Abstract

We show that ridgelets, a system introduced in [E. J. Candes, Appl. Comput. Harmon. Anal., 6 (1999), pp. 197--218], are optimal to represent smooth multivariate functions that may exhibit linear singularities. For instance, let $\{u\cdot x - b >0\}$ be an arbitrary hyperplane and consider the singular function $f(x) = 1_{\{u\cdot x - b > 0\}} g(x)$, where g is compactly supported with finite Sobolev L2 norm $\|g\|_{H^s}$, $s > 0$. The ridgelet coefficient sequence of such an object is as sparse as if f were without singularity, allowing optimal partial reconstructions. For instance, the n-term approximation obtained by keeping the terms corresponding to the n largest coefficients in the ridgelet series achieves a rate of approximation of order n-s/d ; the presence of the singularity does not spoil the quality of the ridgelet approximation. This is unlike all systems currently in use, especially Fourier or wavelet representations.