Amplification factor ($H(\theta )$)

For a one-step translation-invariant recurrence, the amplification factor is the multiplier satisfyingThe Parseval identity converts the pointwise bound $|H|\le 1$ into non-growth of the discrete $2$-norm.

Forthe Fourier convention $\hat{u}(\theta )={\sum }_m{e}^{-im\theta }{u}_m$ givesprovided the denominator does not vanish.