Ridgelets and the representation of mutilated Sobolev functions

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 . x - b> 0} be an arbitrary hyperplane and consider the singular function f (x) = 1 ({u.x-b> 0}) g (x), where g is compactly supported with finite Sobolev L-2 norm parallel tog parallel toH(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.