A generalized uncertainty principle and sparse representation in pairs of bases

An elementary proof of a basic uncertainty principle concerning pairs of representations of R^N vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the improvement of an important result due to Donoho and Huo (2001) concerning the replacement of the l₀ optimization problem by a linear programming (LP) minimization when searching for the unique sparse representation.