Approximation by Refinement Masks

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