Title: $Ψ$ec: A Local Spectral Exterior Calculus,
Abstract: We introduce $\Psi$ec, a local spectral exterior calculus that provides a
discretization of Cartan's exterior calculus of differential forms using
wavelet functions. Our construction consists of differential form wavelets with
flexible directional localization, between fully isotropic and curvelet- and
ridgelet-like, that provide tight frames for the spaces of $k$-forms in
$\mathbb{R}^2$ and $\mathbb{R}^3$. By construction, these wavelets satisfy the
de Rahm co-chain complex, the Hodge decomposition, and that the integral of a
$k+1$-form is a $k$-form. They also enforce Stokes' theorem for differential
forms, and we show that with a finite number of wavelet levels it is most
efficiently approximated using anisotropic curvelet- or ridgelet-like forms.
Our construction is based on the intrinsic geometric properties of the exterior
calculus in the Fourier domain. To reveal these, we extend existing results on
the Fourier transform of differential forms to a frequency domain description
of the exterior calculus, including, for example, a Parseval theorem for forms
and a description of the symbols of all important operators.