This paper develops a mathematical theory of super-resolution, focusing on recovering fine details of an object from coarse-scale information. The authors demonstrate that it is possible to super-resolve point sources with infinite precision by solving a simple convex optimization problem, which can be reformulated as a semidefinite program. This holds as long as 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 the methods are robust to noise. The paper also discusses the stability of super-resolution methods in the presence of noise and provides theoretical results on how the accuracy of the super-resolved signal degrades with increasing noise level and super-resolution factor. Additionally, the authors explore the recovery of piecewise smooth functions and the extension of their results to other types of signals.This paper develops a mathematical theory of super-resolution, focusing on recovering fine details of an object from coarse-scale information. The authors demonstrate that it is possible to super-resolve point sources with infinite precision by solving a simple convex optimization problem, which can be reformulated as a semidefinite program. This holds as long as 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 the methods are robust to noise. The paper also discusses the stability of super-resolution methods in the presence of noise and provides theoretical results on how the accuracy of the super-resolved signal degrades with increasing noise level and super-resolution factor. Additionally, the authors explore the recovery of piecewise smooth functions and the extension of their results to other types of signals.