We present super-algebraic compatible Nyström discretizations for the four Helmholtz boundary operators of Calderón’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 Nyström discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually super-algebraic for smooth solutions.