  Iterative hard thresholding (IHT) and compressive sampling matching pursuit
(CoSaMP) are two types of mainstream compressed sensing algorithms using hard
thresholding operators for signal recovery and approximation. The guaranteed
performance for signal recovery via these algorithms has mainly been analyzed
under the condition that the restricted isometry constant of a sensing matrix,
denoted by $ \delta_K$ (where $K$ is an integer number), is smaller than a
certain threshold value in the interval $(0,1).$ The condition $ \delta_{K}<
\delta^*$ for some constant $ \delta^* \leq 1 $ ensuring the success of signal
recovery with a specific algorithm is called the
restricted-isometry-property-based (RIP-based) bound for guaranteed performance
of the algorithm. At the moment, the best known RIP-based bound for the
guaranteed recovery of $k$-sparse signals via IHT is $\delta_{3k}<
1/\sqrt{3}\approx 0.5774,$ and the bound for guaranteed recovery via CoSaMP is
$\delta_{4k} < 0.4782. $ A fundamental question in this area is whether such
theoretical results can be further improved. The purpose of this paper is to
affirmatively answer this question and rigorously show that the RIP-based
bounds for guaranteed performance of IHT can be significantly improved to $
\delta_{3k} < (\sqrt{5}-1)/2 \approx 0.618, $ and the bound for CoSaMP can be
improved and pushed to $ \delta_{4k}< 0.5102. $ These improvements are achieved
through a deep property of the hard thresholding operator.
