An L∞error bound for the phase approximation problem

Given a random process spectral factor
w(\cdot)
, the phase approximation algorithm, initiated by Jonckheere and Helton [1], constructs a reduced-order spectral factor
\hat{w}(\cdot)
such that
\parallel w/w^{\ast}-\hat{w}/ \hat{w}^{\ast}\parallel
is small in the Hankel-norm sense. In this note, we derive the
L^{\infty}
error bound
\parallel w/w^{\ast} - \hat{w}/\hat{w}^{\ast}\parallel_{\infty} \leq 4(\sigma_{k+1} + ... +\sigma_{N})
, where the σ's are the canonical correlation coefficients. A similar result holds in the multivariable case.