= Positive-real logarithmic amplifier for short arcs
{title2=$\Phi(z)=\sum_j\frac{\delta_j}{D}\frac{1+\delta_j}{1+\delta_j-ze^{-i\theta_j}}$}
Let finitely many arcs have centers $\theta_j$, half-lengths $0<\delta_j\leq1$ and $D=\sum_j\delta_j$. Each summand has positive real part on the closed disk, and $\Phi(0)=1$. On the $j$th arc its contribution has real part at least $1/(4D)$. Thus the analytic logarithm $A=\log\Phi$ obeys $A(0)=0$, $|\operatorname{Im}A|<\pi/2$, and $\operatorname{Re}A\geq\log(1/(4D))$ throughout the arc union. Approximate $A$ by an analytic polynomial and modulate its bounded imaginary part; a partial Fourier sum isolates one half of the polynomial, producing a large value from a bounded function.
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