## A generalized curvelet transform. Approximation properties

**A 2008 Preprint by
F. Blanco-Silva
**

- 2008:06
Some modifications are made to the original de nitions of the three Curvelet Transforms (continuous, semi-discrete and discrete - see [2], [3] and [4]), which improves and simplifies the expressions of the related Parseval-Plancherel formula and Caldeon resolution of the identity. The results presented in this article cast new light on the structure and further properties of curvelet-like schemes of approximation. The discrete curvelet transform obtained here gives rise to a tight frame for the space of square-integrable functions on the plane. Analysis based on manipulation of the corresponding curvelet coefficients (with respect to this frame) helps measure the regularity of functions in different smoothness spaces. This information is used to over characterizations of Lipschitz and Besov spaces, as well as approximation spaces for sequences of finite-dimensional linear spaces spanned by curvelets.