The zero-pinning is a simple yet versatile technique that was recently introduced in the previous paper (2005) for designing FIR orthonormal wavelet filters. The technique basically involves pinning some of the zeros of the parametric Bernstein polynomial to ensure non-negativity of the frequency response. The position of the pinned zeros can be easily adjusted to give a variety of frequency response allowing trade-offs in stopband attenuation and roll-off sharpness. In this paper, we study the effect of the zeros' position on the Sobolev regularity of the equivalent wavelet function