New bounds on the restricted isometry constant δ2k

Restricted isometry constants play an important role in compressed sensing. In the literature, E.J. Candes has proven that delta(2k) < root 2 -1 approximate to 0.4142 is a sufficient condition for the l(1) minimization problem having a k-sparse solution. Later, S. Foucart and M. Lai have improved the condition to delta(2k) < 0.4531 and S. Foucart has improved the bound to delta(2k) < 0.4652. In 2010, T. Cai, L Wang and G. Xu have improved the condition to delta(2k) < 0.4721 for the cases such that k is a multiple of 4 or k is very large and S. Foucart has improved the bound to delta(2k) < 0.4734 for large values of k. In this paper, we have improved the sufficient condition to delta(2k) < 0.4931 for general k. Also, in some special cases, the sufficient condition can be improved to delta(2k) < 0.6569. These new bounds have several benefits on recovering compressible signals with noise. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.