  This paper focuses on the fundamental aspects of super-resolution,
particularly addressing the stability of super-resolution and the estimation of
two-point resolution. Our first major contribution is the introduction of two
location-amplitude identities that characterize the relationships between
locations and amplitudes of true and recovered sources in the one-dimensional
super-resolution problem. These identities facilitate direct derivations of the
super-resolution capabilities for recovering the number, location, and
amplitude of sources, significantly advancing existing estimations to levels of
practical relevance. As a natural extension, we establish the stability of a
specific $l_0$ minimization algorithm in the super-resolution problem.
  The second crucial contribution of this paper is the theoretical proof of a
two-point resolution limit in multi-dimensional spaces. The resolution limit is
expressed as:
  \[
  R = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2}}
\right)}{\Omega}
  \]
  for $\frac{\sigma}{m_{\min}}\leq\frac{1}{2}$, where $\frac{\sigma}{m_{\min}}$
represents the inverse of the signal-to-noise ratio ($\mathrm{SNR}$) and
$\Omega$ is the cutoff frequency. It also demonstrates that for resolving two
point sources, the resolution can exceed the Rayleigh limit
$\frac{\pi}{\Omega}$ when the signal-to-noise ratio (SNR) exceeds $2$.
Moreover, we find a tractable algorithm that achieves the resolution $R$ when
distinguishing two sources.
