Approximation of $L_2(\omega_1,\omega_2)$ Functions by Minimum-Energy Transfer Functions of Linear Systems

The approximation with specified error in $L_2(\omega_1,\omega_2)$ metric of an arbitrary function $F\in L_2(\omega_1,\omega_2)$ by a physically realizable transfer function of a linear system (network) with minimum energy is investigated. The problem is solved on the basis of a spectral decomposition constructed for an integral operator in $L_2(0,\infty)$ with kernel
$$ \frac{\sin\omega_2(t-s)}{\pi(t-s)}-\frac{\sin\omega_1(t-s)}{\pi(t-s)} $$
Secondarily, a criterion is found for a predetermined function $F\in L_2(\omega_1,\omega_2)$ to coincide almost everywhere on $(\omega_1,\omega_2)$ with a certain physically realizable transfer function $G_0$, and a rule is given for reconstructing the function $G_0$ from $F$ in the appropriate complex half-plane.