The Curvelet Transform. A generalized definition and approximation properties.
Abstract
Following ideas from Candès and Donoho in [CD04], [CD05b] and [CD05c], sequences of spaces spanned by curvelets are constructed. Some modifications are made to the original definitions of the three Curvelet Transform cases (continuous, semi-discrete, discrete), which improves and simplifies the expressions of the related Parseval-Plancherel formula and Calderón resolution of the identity. The results presented in this dissertation cast new light on the structure and further properties of curvelets, and offer means to extend such construction to higher dimensions. The obtained discrete curvelet transform 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 offer characterizations of Lipschitz and Besov spaces, as well as approximation spaces for sequences of finite-dimensional linear spaces spanned by curvelets.
Degree
Ph.D.
Advisors
Lucier, Purdue University.
Subject Area
Mathematics
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