Convergence of wavelet frame operators as the sampling density tends to infinity

In this paper, we study the convergence of wavelet frame operators defined by Riemann sums of inverse wavelet transforms. We show that as the sampling density tends to the infinity, the wavelet frame operator tends to the identity or embedding mapping in various operator norms provided the wavelet function satisfies some smoothness and decay conditions. As a consequence, we also get some spanning results of affine systems.