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.

  • 1
    $\begingroup$ Yes. Replace the Fourier modes by the eigenfunctions of the Dirichlet problem. $\endgroup$ Commented Dec 14, 2017 at 23:01
  • $\begingroup$ If you do as Michael rightfully suggests, the zero-mean condition will be replaced with the vanishing-at-the-boundary condition. $\endgroup$ Commented Dec 15, 2017 at 10:29
  • $\begingroup$ @MichaelRenardy do you mean Dirichlet Laplacian problem ? $\endgroup$ Commented Dec 15, 2017 at 15:37
  • $\begingroup$ Yes. Blahblahblah blah (added because they want at least 15 characters). $\endgroup$ Commented Dec 15, 2017 at 23:34
  • $\begingroup$ @GiuseppeNegro What should vanish at boundary ? The function f ? Could you explain the reasoning ? $\endgroup$ Commented Dec 20, 2017 at 20:20


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