Sharp Jackson and Bernstein inequalities for $N$-term approximation in sequence spaces with applications Gustavo GarrigosEugenio Hernandez 41A1742C40non-linear approximationanisotropic Besov and Triebel-Lizorkin spaces$phi$-transformswavelets We study $N$-term approximation for general families of sequence spaces, establishing sharp versions of Jackson and Bernstein inequalities. The sequence spaces used are adapted to provide characterizations of Triebel-Lizorkin and Besov spaces by means of wavelet-like systems using general dilation matrices, and thus they include spaces of anisotropic smoothness. As an application, we characterize the $N$-term approximation spaces when the error is measured in the first of the spaces mentioned above. Indiana University Mathematics Journal 2004 text pdf 10.1512/iumj.2004.53.2552 10.1512/iumj.2004.53.2552 en Indiana Univ. Math. J. 53 (2004) 1741 - 1764 state-of-the-art mathematics http://iumj.org/access/