The paper by Emmanuel Candès and Justin Romberg addresses the problem of reconstructing a sparse signal \( x^0 \in \mathbb{R}^n \) from a limited number of linear measurements. Given \( m \) randomly selected samples of \( Ux^0 \), where \( U \) is an orthonormal matrix, they show that \( \ell_1 \)-minimization can recover \( x^0 \) exactly when the number of measurements exceeds a certain threshold. This threshold is given by: \[ m \geq \text{Const} \cdot \mu^2(U) \cdot S \cdot \log n, \] where \( S \) is the number of nonzero components in \( x^0 \), and \( \mu \) is the largest entry in \( U \) properly normalized. The result holds for "most" sparse signals \( x^0 \) supported on a fixed set \( T \). Given \( T \), if the sign of \( x^0 \) for each nonzero entry on \( T \) and the observed values of \( Ux^0 \) are drawn at random, the signal is recovered with overwhelming probability. The paper also discusses the relevance of the parameter \( \mu(U) \) and provides numerical experiments to illustrate the effectiveness of the method in various applications, such as wavelet subband sampling and noiselet measurements. The main theorem and its proof are detailed, showing that the recovery is possible under specific conditions on the measurement matrix \( U \) and the signal support \( T \).The paper by Emmanuel Candès and Justin Romberg addresses the problem of reconstructing a sparse signal \( x^0 \in \mathbb{R}^n \) from a limited number of linear measurements. Given \( m \) randomly selected samples of \( Ux^0 \), where \( U \) is an orthonormal matrix, they show that \( \ell_1 \)-minimization can recover \( x^0 \) exactly when the number of measurements exceeds a certain threshold. This threshold is given by: \[ m \geq \text{Const} \cdot \mu^2(U) \cdot S \cdot \log n, \] where \( S \) is the number of nonzero components in \( x^0 \), and \( \mu \) is the largest entry in \( U \) properly normalized. The result holds for "most" sparse signals \( x^0 \) supported on a fixed set \( T \). Given \( T \), if the sign of \( x^0 \) for each nonzero entry on \( T \) and the observed values of \( Ux^0 \) are drawn at random, the signal is recovered with overwhelming probability. The paper also discusses the relevance of the parameter \( \mu(U) \) and provides numerical experiments to illustrate the effectiveness of the method in various applications, such as wavelet subband sampling and noiselet measurements. The main theorem and its proof are detailed, showing that the recovery is possible under specific conditions on the measurement matrix \( U \) and the signal support \( T \).