A mathematical theory of super-resolution and diffraction limit

This paper is devoted to elucidating the essence of super-resolution and deals mainly with the stability of super-resolution and the diffraction limit. The first discovery is two location-amplitude identities characterizing the relations between source locations and amplitudes in the super-resolution problem. These identities allow us to directly derive the super-resolution capability for number, location, and amplitude recovery in the super-resolution problem and improve state-of-the-art estimations to an unprecedented level to have practical significance. The nonlinear inverse problems studied in this paper are known to be very challenging and have only been partially solved in recent years, but we now have a clear and simple picture of all of these problems, which allows us to solve them in a unified way in just a few pages. The second crucial result of this paper is the theoretical proof of a two-point diffraction limit in spaces of general dimensionality under only an assumption on the noise level. The two-point diffraction limit is given by \[ \mathcal{R} = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2}} \right)}{\Omega} \] for $\frac{\sigma}{m_{\min}}\leq\frac{1}{2}$, where $\frac{\sigma}{m_{\min}}$ represents the inverse of the signal-to-noise ratio ($SNR$) and $\Omega$ is the cutoff frequency. In the case when $\frac{\sigma}{m_{\min}}>\frac{1}{2}$, there is no super-resolution in certain cases. This solves the long-standing puzzle and debate about the diffraction limit for imaging (and line spectral estimation) in very general circumstances. Our results also show that, for the resolution of any two point sources, when $SNR>2$, one can definitely exceed the Rayleigh limit $\frac{\pi}{\Omega}$, which is far beyond common sense. We also find the optimal algorithm that achieves the optimal resolution when distinguishing two sources.