The optimum approximation of signals having quite small side-lobes

We consider set of signals with a main-lobe and a pair of small side-lobes. Weighted square-integral of the main-lobe is assumed to be bounded. Moreover, in order to define divergence of side-lobes of error in the sense of worst-case amplitude in attenuation band, we introduce a measure like Kullback-Leibler divergence and this measure is assumed to be bounded. We prove that the presented approximation minimizes various measures of error of running approximation at the same time.