We introduce $\mathrm{\Psi }\mathrm{ec}({\mathbb{R}}^n)$, 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 ${\mathrm{\Omega }}^r({\mathbb{R}}^n)$ 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 $\mathrm{\Psi }\mathrm{ec}({\mathbb{R}}^n)$ 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.

我们介绍 $\mathrm{\Psi }\mathrm{ec}（{\mathbb{[R}}^{ñ}）$，使用小波将Cartan的微分形式的外部演算离散化。我们的构造由具有灵活方向性定位的差分r形小波组成，可为空间提供紧密的框架${\mathrm{\Omega }}^{\mathrm{[R}}（{\mathbb{[R}}^{ñ}）$ 的形式 ${\mathbb{[R}}^2$ 和 ${\mathbb{[R}}^3$。通过构造，小波满足de Rahm共链复数，Hodge分解，并且r形式的k维积分为$（\mathrm{[R}-ķ）$-形成。他们还验证了微分形式的斯托克斯定理，其中最有效的有限维近似是使用定向局部化，曲波状或脊波状形式实现的。的建设$\mathrm{\Psi }\mathrm{ec}（{\mathbb{[R}}^{ñ}）$建立在傅立叶域中外部演算的几何简单性上。我们通过将微分形式的傅立叶变换上的现有结果扩展到外部演算的频率描述来建立这种结构，例如，包括形式的Plancherel定理和所有重要算符符号的描述。