Weighted Hankel-norm approximation: Calculation of bounds

In [I], a method was given for optimal Hankel-norm approximation with frequency weighting. In a broad sense, this method stands in the same relation to frequency-weighted balanced approximation [2] as unweighted optimal Hankel-norm approximation stands in relation to unweighted balanced approximation, see, e.g. [3]. No bounds were given in [I] on the frequency-domain error associated with frequency-weighted optimal Hankel-norm approximation; the derivation and statement of such bounds is the main task of the present work. In [3], frequency-domain bounds are given for unweighted optimal Hankel-norm approximations. The bounds given here for the weighted case essentially are obtained by introducing to the bounds of [3] an extra multiplicative factor which depends on the weighting function, but not on the object being approximated. This extra factor will be unity if the weighting function is constant. (Thus, as one would hope, the unweighted results are obtained.) The extra factor will tend to be large when there is significant variation between the maximum and minimum amplitudes of the weighting function along the frequency axis.