Spectral Roll-Off

Definition: A function $W(\omega)$ is said to be of order $1/\omega^{n+1}$ if there exist $\omega_0$ and some positive constant $M<\infty$ such that $\left\vert W(\omega)\right\vert<M/w^{n+1}$ for all $\omega > \omega_0$ .

Theorem: (Riemann Lemma): If the derivatives up to order $n$ of a function $w(t)$ exist and are of bounded variation, then its Fourier Transform $W(\omega)$ is asymptotically of order $1/\omega^{n+1}$ , i.e.,

$$W(\omega) = {\cal O}\left(\frac{1}{\omega^{n+1}}\right), \quad(\hbox{as }\omega\to\infty)$$ (3.42)

Proof: See §B.18.