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 · x - b >0\ be an arbitrary hyperplane and consider the singular function f(x) = 1_· x - b > 0\ g(x), where g is compactly supported with finite Sobolev L2 norm \|g\|Hˢ, $s > 0$. 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.