On the Theorem of Uniform Recovery of Random Sampling Matrices

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling ma-trices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m & s(ln s)3 lnN. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m×N-matrices, by considering what we call low entropy. We also present an improved condition on the so-called restricted isometry constants, δs, ensuring sparse recovery via ` 1-minimization. We show that δ2s < 4/ 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type δ2s < 2/3