Prony series

Remarks

Example: the standard linear solid has $G(t) = E_\infty + E_1 e^{-t/\tau}$, a Prony series with $n = 1$, $G_\infty = E_\infty$ and $G_1 = E_1$. Non-example: the power law $G(t) = C t^{-\alpha}$ with $C > 0$ and $0 < \alpha < 1$ is not a Prony series, because every Prony series is bounded near $t = 0$ while $t^{-\alpha}$ is not. Such a power law is a continuous superposition of exponentials, $t^{-\alpha} = \frac{1}{\Gamma(\alpha)} \int_0^\infty u^{\alpha - 1} e^{-ut} \, du$, and Prony series approximate it on a bounded range of times. In the Laplace domain, $s \hat G(s) = G_\infty + \sum_i G_i \tau_i s / (1 + \tau_i s)$.