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.