= Complementary transition bands for a scaling function
{title2=$f(t)^2+f(t-2\pi)^2=1$}
Let a real measurable <Fourier transform> equal one on $|t|<2\pi/3$, vanish on $|t|\ge4\pi/3$, and satisfy the displayed complementary identity on the positive transition band. On a fundamental interval, either only the central translate is nonzero, or exactly the two complementary translates contribute. Thus <orthonormal translates and Fourier periodization> holds. The periodic mask defined on $[-\pi,\pi]$ by $m(t)=f(2t)$ for $|t|<2\pi/3$ and $m(t)=0$ otherwise satisfies the <scaling refinement equation> in Fourier form. A concrete even transition is $f(t)=\cos(\pi(3|t|/(2\pi)-1)/2)$ on the transition bands. For complex-valued $f$, ordinary squares cannot replace absolute squares: transition values $i$ and $\sqrt2$ have squares summing to one but squared moduli summing to three.
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