Dynamical System State Need not Have Spectrum

A classification of LSI polynomials into optimal and suboptimal LSI polynomials is presented in this note. By an example it is shown that a suboptimal LSI polynomial can be unstable even if the original polynomial does not have any zeros on the unit circle. This has invalidated the proof for modified Shanks’ conjecture in the 2-D case as presented in the paper’ leaving the conjecture to remain a conjecture. We also hope that this note will clarify the well-known Robinson’s result regarding the stabilization using least squares approach.