$\Psi$ec: A Local Spectral Exterior Calculus

Abstract

We introduce $\Psi \mathrm{ec}$, a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential $r$-form wavelets with flexible directional localization that provide tight frames for the spaces ${\Omega }^r({\mathbb{R}}^n)$ of forms in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the $k$-dimensional integral of an $r$-form is an $(r-k)$-form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of $\Psi \mathrm{ec}$ builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.

Revised version, updated figures

References in corpus (2)

• The Geometry of Supermanifolds and New Supersymmetric Actions
• A Local Fourier Slice Theorem

Cited by in corpus (1)

• Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes