Approximation by Refinement Masks

We construct a Parseval wavelet frame with compact support for an arbitrary continuous \(2\pi\) -periodic function \(f\) , \(f(0)=1\) , satisfying the inequality \(|f(x)|^2+|f(x+\pi)|^2\le 1\) . The frame refinement mask uniformly approximates \(f\) . The refining function has stable integer shifts.