  We present superalgebraic compatible Nystr\"om discretizations for the four
Helmholtz boundary operators of Calder\'{o}n's calculus on smooth closed curves
in 2D. These discretizations are based on appropriate splitting of the kernels
combined with very accurate product-quadrature rules for the different
singularities that such kernels present. A Fourier based analysis shows that
the four discrete operators converge to the continuous ones in appropriate
Sobolev norms. This proves that Nystr\"om discretizations of many popular
integral equation formulations for Helmholtz equations are stable and
convergent. The convergence is actually superalgebraic for smooth solutions.
