This paper presents a mathematical theory of super-resolution, demonstrating that it is possible to recover the exact locations and amplitudes of point sources from low-frequency Fourier samples. The key idea is to use a convex optimization problem, specifically a total-variation minimization, which can be reformulated as a semidefinite program. The method is robust to noise and works under the condition that the distance between sources is at least $ 2/f_c $, where $ f_c $ is the frequency cut-off. The theory extends to higher dimensions and other models, and it is shown that the method can recover piecewise smooth functions by resolving their discontinuities with infinite precision. The paper also discusses the relationship between the super-resolution factor and the accuracy of the recovered signal, and provides theoretical results explaining how the accuracy degrades with increasing noise and super-resolution factor. The results are validated through numerical simulations and are shown to be tight in certain cases. The paper also compares the proposed method with related work in the field of sparse recovery and highlights the advantages of the total-variation approach in terms of robustness and accuracy.This paper presents a mathematical theory of super-resolution, demonstrating that it is possible to recover the exact locations and amplitudes of point sources from low-frequency Fourier samples. The key idea is to use a convex optimization problem, specifically a total-variation minimization, which can be reformulated as a semidefinite program. The method is robust to noise and works under the condition that the distance between sources is at least $ 2/f_c $, where $ f_c $ is the frequency cut-off. The theory extends to higher dimensions and other models, and it is shown that the method can recover piecewise smooth functions by resolving their discontinuities with infinite precision. The paper also discusses the relationship between the super-resolution factor and the accuracy of the recovered signal, and provides theoretical results explaining how the accuracy degrades with increasing noise and super-resolution factor. The results are validated through numerical simulations and are shown to be tight in certain cases. The paper also compares the proposed method with related work in the field of sparse recovery and highlights the advantages of the total-variation approach in terms of robustness and accuracy.