The usual formulation of $H^{-1}$ norm for a zero-mean periodic function on some domain $\Omega\in\mathbb{R}$ is as follows:

$\|f\|^2_{H^{-1}}=\sum\limits_{k\in Z, k\neq 0}\dfrac{\hat{f}^2_k}{k^2}$, where $\hat{f}_k$ is the $k$-th fourier coefficient.

Is it possible to formulate something similar for a non-zero mean non-periodic function on a bounded domain.

6more comments