 Uniform power bound from separated amplification roots

ID: uniform-power-bound-from-separated-amplification-roots

For a uniformly bounded two-level Fourier companion matrix

C (θ)

with distinct roots

g_{1}, g_{2}

of modulus at most one, its powers satisfy

C^{n} = \frac{g _{1}^{n} ( C - g _{2} I ) - g _{2}^{n} ( C - g _{1} I )}{g _{1} - g _{2}} .

A positive frequency-uniform gap therefore gives a uniform bound on every power. The Parseval identity then supplies spatial L2 norm stability. When the gap closes at a repeated unit root, a Jordan block can produce growth proportional to the number of steps despite both roots having modulus one.

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