Ec: a Local Spectral Exterior Calculus

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Ψec: A Local Spectral Exterior Calculus

Christian Lessig1 Otto-von-Guericke-Universit¨atMagdeburg

Abstract

We introduce Ψec(Rn), 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 Ωr(Rn) of forms in R2 and R3. 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 Ψec(Rn) 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. Keywords: exterior calculus, wavelets, structure preserving discretization 2010 MSC: 42C15, 42C40, 65T60, 58A10

1. Introduction

The exterior calculus, first introduced by Cartan [1], provides a formulation of scalar and vector-valued functions that encodes their differential relation- ships in a coordinate-invariant language. The central objects of the calculus are differential forms, which provide a covariant formalization of vector fields, and the exterior derivative, which is a first order differential operator generalizing

arXiv:1811.12269v3 [cs.NA] 5 Oct 2020 gradient, curl, and divergence. The importance of the exterior calculus for the numerical solution of partial differential equations, even in Rn and with a flat metric, has been shown in various applications. One of the first ones was electromagnetic theory where already in the 1960s it was realized [2] that the electric and magnetic fields have to be discretized differently to obtain satisfactory numerics. Later, it was found that this could be understood by the electric and magnetic fields being differential forms of different degrees [3, 4]. Motivated by these and analogous

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Preprint submitted to Applied and Computational Harmonic Analysis October 7, 2020 results in other fields, e.g. in geophysical fluid dynamics [5, 6, 7], various discrete realizations of Cartan’s exterior calculus have been proposed, e.g. [8, 9, 10, 11, 12, 13]. These have been applied to applications such as elasticity [10], fluid mechanics [14, 15], climate simulation [7, 11, 16], magnetohydrodynamics [17], electromagnetic theory [3], and geometry processing [18]. In the present work, we propose Ψec(Rn), an alternative discretization of Cartan’s exterior calculus that is defined over a family of differential form wavelets with flexible directional localization. Towards its construction, we extend existing results from the theoretical physics literature and develop a description of the exterior calculus in the Fourier domain. This shows that under the Fourier transform it becomes a chain complex whose structure is encoded in a simple, geometric way in spherical coordinates in frequency space. The exterior derivative, for instance, acts only on the radial component of a differential form in frequency space. We therefore discretize the exterior calculus in the Fourier domain using spherical coordinates and attain space-frequency localization by using polar wavelet windows [19, 20] that respect the spherical coordinate structure. The construction yields differential form wavelets well localized in space and frequency that obey by construction the de Rahm complex and the distinction between exact and co-exact forms, i.e. the Hodge-Helmholtz decomposition. At the same time, the definition in spherical coordinates also allows for a flexible directional localization, between fully isotropic and curvelet- and ridgelet-like. In contrast to most existing discretizations in the literature, the “discrete” differential form wavelets in our work are bona fide forms in the sense of the continuous theory. Hence, all operations from the continuous setting are well defined. A central question therefore becomes efficient numerical evaluation and we will at least partially address it in the present work. n r,ν,n The properties of Ψec(R ) and its differential form wavelets ψs,a , where n = 2, 3 is the dimension of the space, 0 ≤ r ≤ n is the degree of the form, ν ∈ {d, δ} denotes if the wavelet is exact or co-exact, and s = (js, ks, ts) describes level js, translation ks and orientation ts, can be summarized as follows (see Sec. 5 for the precise technical statements):

r,ν,n ˙ |ν| r n • The ψs,a form tight frames for the homogeneous Sobolev spaces L2 (Ων , R ). • The diﬀerential form wavelets satisfy the exterior co-chain complex, i.e. r,δ,n r+1,d,n r,d,n dψs,a = ψs,a and dψs,a = 0. Furthermore, the exterior derivative d preserves the localization described by s.

• Stokes’ theorem for diﬀerential forms α ∈ Ωr(Rn), Z Z α = dα, ∂M M

for M ⊂ Rn a manifold, holds for diﬀerential form wavelets as Z Z X r−1,δ X r,d αs ψs ∧ χ∂M = αs ψs ∧ χM (1) n n s R s R

2 r n where the αs are the frame coeﬃcients of α ∈ Ω (R ) in the diﬀerential

form wavelet frame, and χM and χ∂M are the characteristic functions of M and its boundary ∂M, respectively. When dim(M) = n, Stokes theorem therefore contains the approximation problem for cartoon-like functions (right hand side of Eq. 1). This has been studied extensively in the literature on curvelets, shearlets and related constructions, e.g. [21, 22, 23, 24], and these provide for it quasi-optimal approximation rates.

Moreover, χ∂M is exactly the wavefront set associated with χM .

k,ν,n • The Hodge dual ? ψs,a has a simple and practical description. r,ν,n • The ﬁber integral of ψs,a along an arbitrary p-dimensional linear sub- r−p,ν,n 0 manifold is again a diﬀerential form wavelet ψs0,a and with s preserving the localization of s.

• The Laplace-de Rahm operator ∆ : Ωk → Ωk has a closed form represen- r,ν,n tation and Galerkin projection for the differential form wavelets ψs,a . r,ν,n • The ψs,a have closed form expressions in the spatial domain. This clar- ifies, e.g. how the angular localization in frequency space is carried over to the spatial domain. Multiplicative operators in the exterior calculus, such as the wedge product or the Lie derivative, have currently no natural expression in our calculus. In numerical calculations, these can be determined using the transform method [25, 26], i.e. with the multiplication being evaluated pointwise in space, which is facilitated by the closed form expressions that are available there. A thorough investigation of this has, however, be left for future work. The remainder of the work is structured as follows. After discussing related work in the next section, we present in Sec. 3 some background on the polar wavelets that we use as localization windows in the frequency domain. Afterwards we develop in Sec. 4 the Fourier transform of differential forms. In Sec. 5 we introduce our differential form wavelets and study how the exterior calculus, including the exterior derivative, the wedge product, Hodge dual, and Stokes theorem, can be expressed using these. In Sec. 6 we summarize our work and discuss directions for future work. Our notation and conventions are summarized in Appendix A.

2. Related Work

Our work builds on ideas from what are usually two distinct fields: firstly, wavelet theory and in particular wavelets constructed in polar and spherical coordinates, and, secondly, exterior calculus and its discretizations. We will discuss relevant literature in these fields in order.

3 Polar Wavelets. The idea to combine space-frequency localization with a description in polar coordinates in the Fourier domain goes back at least to work by Fefferman in the theory of partial differential equations, in particular on the second dyadic decomposition, cf. [27, Ch. IX]. It was re-introduced multiple times, e.g. in work on steerable wavelets [28, 29] and curvelets [30, 21, 22] where the motivation was the analysis of directional features in images. A framework that encompasses the latter two approaches was proposed by Unser and co-workers [31, 32, 19, 33] and this defines the polar wavelets that are the scalar building block for the construction of Ψec(Rn). We use the term ‘polar wavelet’, or short ‘polarlet’, since it encapsulates what distinguishes the construction from other multi-dimensional wavelets and what is the key to their utility in our work.

Curl and Divergence Free Wavelets. Various constructions for curl and divergence free wavelets have been proposed over the years, e.g. [34, 35, 36, 37, 38, 39], and these are related to our differential form wavelets by the musical isomorphism. Based on seminal work by Lemairé-Rieusset[34], various authors, e.g. [40, 41, 42, 43, 44] constructed coupled scalar wavelet bases for 1-dimensional domains that satisfy a differential relationship so that the derivative of one yields (up to a constant) the other or a corresponding integral relationship holds. These constructions can be extended to curl or divergence free wavelets using the coordinate representations of the differential operators of vector calculus. Our differential form wavelets satisfy a similar coupling under differentiation. However, our construction is inherently multi-dimensional and we exploit the simple structure of the differential operators in the frequency domain. The latter has also been used in the work by Bostan, Unser and Ward [39] although divergence freedom is there only enforced numerically. A precursor to the presented work is those in [45]. There, polar wavelets are used as scalar window functions for the construction of polar divergence free wavelets and it is exploited that a vector field is divergence free when its Fourier transform is tangential to the frequency sphere.

Discretizations of Exterior Calculus. Discretizations of Cartan’s exterior calculus try to preserve important structures of the continuous theory, e.g. the de Rahm co-chain complex or Stokes’ theorem, to improve the accuracy and robustness of numerical calculations. Different precursors those good numerical performance can, at least in hindsight, be explained through their adherence to the exterior calculus, exist. For example, spectral discretizations in geophysical fluid dynamics, first introduced in the 1950s and 1960s, yield energy and enstrophy conserving time integrators [5] by implicitly using what one could call a spectral exterior calculus [46]. In the 1970s, Arakawa and Lamb [6, 7] introduced the now eponymous grid-based discretizations for geophysical fluid dynamics to extend energy and enstrophy conservation to finite difference methods, again without an explicit connection to Cartan’s calculus but implicitly respecting its structure. A few years later, Nédélec[47, 48], introduced consistent discretizations for the spaces H(curl) and H(div) that respect important properties of the

4 classical differential operators of vector calculus, and hence also of the exterior derivative. The first work that, to our knowledge, made an explicit connection between exterior calculus and numerics was those by Bossavit [3] in computational electromagnetism who realized that the use of differential forms and the calculus defined on them enables one to systematically understand and extend earlier results that provided good numerical performance for problems in computational electromagnetism [2]. Most existing discretizations of exterior calculus are finite element-based, such as finite element exterior calculus developed by Arnold and collabora- tors [10, 49], discrete exterior calculus by Hirani, Desbrun and co-workers [50, 9], mimetic discretizations by Bochev and Hyman [8], and the TRiSK scheme by Thuburn, Ringler, Klemp, and Skamarock [51, 11]. These works all have in common that they construct a discrete analogue of Cartan’s exterior calculus that satisfies important structural properties of the continuous theory, e.g. in that it is a co-chain complex for a discrete exterior derivative or that a discrete Stokes’ theorem holds. Our construction, in contrast, is best seen as a multi-resolution projection of the continuous theory that preserve its intrinsic structure. In particular, our differential form wavelets remain genuine forms in the sense of the continuous theory. All operators from there are thus naturally defined. Compared to finite element type discretizations where one seeks discrete definitions mimicking the original theory, the questions shifts in our work thus to how these can be computed efficiently numerically or how this can be ensured in the construction of the wavelets. Wavelets and exterior calculus have so far only been combined in two works and again within a finite element framework. Dubos, Kevlahan and collabora- tors [16, 52, 53] extended the TRiSK scheme to a multi-resolution formulation using second generation wavelets and demonstrated that this provides an efficient discretization for geophysical fluid dynamics. In work contemporaneous to ours, Budninskiy, Owhadi, and Desbrun [54] constructed operator-adapted wavelets for Discrete Exterior Calculus using the efficient algorithms proposed in [55]. In contrast to Ψec(Rn) whose construction relies on the structure of exterior calculus in the Fourier domain, the above works are defined purely in the spatial domain. The extension of finite element-based discretizations of exterior calculus to spectral convergence rates has been considered by Rufat, Mason and Des- brun [12]. These authors accomplished it by introducing interpolation and histapolation maps to relate discrete exterior calculus and continuous forms. Gross and Atzberger presented a similar approach using hyper-interpolation for radial manifolds, i.e. manifolds that can be described as a height field over S2. A discretizations of exterior calculus for subdivision surfaces was introduced by de Goes et al. [13]. It shares the multi-resolution structure with our approach and, at least in the limit case, also yields continuous differential forms. Another recent discretization of exterior calculus is those by Berry and Gi- annakis [56] for manifold learning problems. They employ the eigenfunctions of the Laplace-de Rahm operator as bases for their forms. This is analogous to the spectral bases, formed by spherical harmonics, first used in geophysical

5 ﬂuid dynamics and that, as mentioned earlier, essentially form a spectral exterior calculus. We also build on a spectral formulation but localize it in space and frequency and, furthermore, exploit the additional structure aﬀorded in the Euclidean setting.

3. Polar Wavelets

Polar wavelets are a family of wavelets deﬁned in polar or spherical coordinates in the Fourier domain. Their construction uses a compactly supported radial window, hˆ(|ξ|), which controls the overall frequency localization, and an angular one,γ ˆ(ξ¯), which controls the directionality, where ξ¯ = ξ/|ξ|. The mother wavelet is thus given by ψˆ(ξ) = ψˆ(ξ¯) hˆ(|ξ|) with the whole family of functions being generated by dilation, translation and rotation. In two dimensions, the angular window can be described with a Fourier series in the polar angle θˆ. A polar wavelet is there hence given by

j Nj 2 X ˆ j ψˆ (ξ) ≡ ψˆ (ξ) = βj,t einθ hˆ(2−j|ξ|) e−ihξ,2 ki (2) s jkt 2π n n=−Nj where the multi-index s = (j, k, t) describes level j, translation k, and orientation t and we will write (js, ks, ts) whenever confusion might arise. The coeﬃ- j,t j,t cients βn control the angular localization and in the simplest case βn = δn0 for all j, t and one has isotropic, bump-like wavelet functions. Conversely, when the j,t support of the βn are the entire integers in n, then one can describe compactly supported angular windows. The above formulation then also encompasses second generation curvelets [22] and provides more generally a framework to realize α-molecule-like constructions [57, 58, 59]. A useful property of polar wavelets is that their spatial representation, given by inverse Fourier transform F−1 of Eq. 2, can be computed in closed form. Using the Fourier transform in polar coordinates, cf. Appendix A.4, one obtains [20]

Nj 2j X ψ (x) ≡ ψ (x) = in βj,t einθx h (|2jx − k|) (3) s jkt 2π n n n=−Nj ˆ ˆ where hn(|x|) is the Hankel transform of h(|ξ|) of order n. When h(|ξ|) is the window proposed for the steerable pyramid [60], then hn(|x|) also has a closed form expression [20]. Note also that the angular localization described by the j,t βn is essentially invariant under the Fourier transform and only modiﬁed by the factor of in that implements a rotation by π/2. For applications, the wavelets in Eq. 3 are augmented using scaling functions φj,k. They are isotropic and deﬁned using a radial windowg ˆ(|ξ|), i.e. φj,0 = −1 F (ˆg), that is chosen such that the φj,k represent a signal’s low frequency part. To simplify notation, we follow the usual convention and write ψ−1,k ≡ φ0,k. With the scaling functions, the following result holds [19, 45].

6 Tj ×2Nj +1 Proposition 1. Let Uj ∈ R be the matrix formed by the angular local- j,t j −int(2π/Tj ) ization coefficients βn = βn e for the Tj different orientations, and 2Nj +1×2Nj +1 let Dj ∈ R be a diagonal matrix. When the Calderónadmissibility 2 P∞ ˆ −j 2 2 H condition gˆ(|ξ|) + j=0 h(2 |ξ|) = 1, ∀ξ ∈ Rb , is satisfied and Uj Uj = Dj 2 with tr(Dj) = 1 for all levels j, then any f ∈ L2(R ) has the representation

∞ Tj X X X f(x) = f(y), ψj,k,t(y) ψj,k,t(x) 2 j=−1 k∈Z t=1 with frame functions

2j ψ (x) = ψR (2jx − k) j,k,t 2π 2πt/Tj

where ψ(x) is given by Eq. 3 and R2πt/Tj is the rotation by 2πt/Tj. Although the above frame is redundant, since it is Parseval tight it still aﬀords most of the conveniences of an orthonormal basis. In the following, we will often write S for the index set of all scales, translations, and rotations. In three dimensions, polar wavelets are deﬁned analogous to Eq. 2 by

3j/2 Lj l 2 X X j ψˆ (ξ) ≡ ψˆ (ξ) = κj,t y (ξ¯) hˆ(2−j|ξ|) e−ihξ,2 ki (4) s j,k,t (2π)3/2 lm lm l=0 m=−l where the directional localization windowγ ˆ(ξ¯) is now expressed using the spher- j,t ical harmonics ylm and the κlm are the expansion coeﬃcients; see Appendix A.3 for a brief summary of harmonic analysis of the sphere. The wavelets have again a closed form expression in the spatial domain,

Lj l 23j/2 X X ψ (x) ≡ ψ (x) = ilκj,t y (¯x) h (|2jx − k|). (5) s j,k,t (2π)3/2 lm lm l l=0 m=−l which can be obtained using the Rayleigh formula, cf. Appendix A.4. Further- more, the wavelets can again be augmented using scaling functions to represent a signal’s low frequency part. Then the following result holds [33, 45] (see again Appendix A.3 for notation).

2 (Lj +1) Proposition 2. Let wj,t ∈ R be the vector formed by the rotated angular j,t Pl m0 localization coefficients κlm = m0=−l W (λj,t)l,m κl,m0 for a localization win- m dow centered at λj,t, where W (λj,t)l,m is the Wigner-D matrix implementing 2 2 rotation in the spherical harmonics domain, and let Glm ∈ R(Lj +1) ×(Lj +1) be the matrix formed by the spherical harmonics product coefficients for fixed 2 P∞ ˆ −j 2 3 (l, m). When the Calderóncondition gˆ(|ξ|) + j=0 h(2 |ξ|) = 1, ∀ξ ∈ Rb PTj lm 3 is satisfied and δl,0δm,0 = t=0 wj,t G wj,t for all j, t then any f(x) ∈ L2(R )

7 has the representation

∞ Tj X X X f(x) = f(y) , ψj,k,t(y) ψj,k,t(x) 3 j=−1 k∈Z t=1 with frame functions

23j/2 ψ (x) = ψR (2jx − k), j,k,t (2π)3/2 λj,t

for ψ(x) deﬁned in Eq. 5 and Rλj,t the rotation from the North pole to λj,t. See Appendix B for a proof of the admissibility conditions. We refer to the original works [19, 33] and [20] for a more detailed discussion of polar wavelets.

4. Diﬀerential Forms and their Fourier Transform

In this section, we first recall basic facts about differential forms and the exterior calculus defined on them. Our principle reference for this material will be the book by Marsden, Ratiu, and Abraham [61] and we will also use the notation and conventions from there. Afterwards, starting from existing results in theoretical physics [62, 63, 64], we will define and study the Fourier transform of the exterior calculus.

4.1. Tangent and Cotangent Bundle The setting of our discussion in the following will be Rn with n = {2, 3}. Although the tangent bundle T Rn can be identiﬁed with Rn × Rn, for our work it is of importance to consistently distinguish Rn and the tangent and cotangent bundles, T Rn and T ∗Rn, respectively. We will denote a generic basis for the tangent space as

∂ ∂ span , ··· , = T n ∂x1 ∂xn xR

i n with the ∂/∂x in general being dependent on the fiber TxR they span (such as in spherical coordinates). The ∂/∂xi induce a dual basis dxi for the cotangent ∗ n i j bundle T R through the fiberwise duality condition dx ∂/∂x = δij. We will also typically use the Einstein summation convention over repeated indices and hence employ “upstairs” indices for components of contra-variant vectors n ∗ n in TxR and “downstairs” ones for covariant vectors in Tx R . For example, a vector field X ∈ X(Rn) is in coordinates given by Xi ∂/∂xi and a co-vector field i by αi dx (summation implied).

8 4.2. Differential Forms and the Exterior Calculus A differential r-form α ∈ Ωr(Rn) on Rn, with 0 ≤ r ≤ n, is a covariant, anti- symmetric tensor field of rank r, i.e. an anti-symmetric, multi-linear functional on the rth power of the tangent bundle. When r = 0 this corresponds to functions, i.e. Ω0(Rn) =∼ F(Rn), and for r = n one has densities, e.g. f(x) dx. All other degrees r correspond to “vector-valued” fields. For example, a 1-form is a co-vector field, i.e. a section of the cotangent bundle T ∗Rn, and since it is a functional on T Rn it naturally pairs with the tangent vector field of a curve. A 2-form, correspondingly, pairs with the tangent vectors of a 2-manifold and in R3 it can hence be thought of as an infinitesimal area. The result of the pairing between an r-form and the tangent vectors of an r-manifold can be integrated and one thus also says that an r-form is to be integrated over an r-manifold [9, 65]. A principal motivation for studying differential forms is that they are closed under the exterior derivative d,

r n r+1 n d : Ω (R ) → Ω (R ) (6) which provides a coordinate invariant derivative for tensor fields. The exterior derivative furthermore satisfies d ◦ d = 0. The differential forms under d hence form a co-chain complex known as the de Rahm complex,

d 0 n d d n n d 0 −−→ Ω (R ) −−→· · · −−→ Ω (R ) −−→ 0.

When a differential form α ∈ Ωr(Rn) satisfies dα = 0 then α is closed and when there exists a β such that β = dα then α is exact. The exterior calculus also provides a product on differential forms known as the wedge product

∧ :Ωr × Ωl → Ωr+l. (7)

It turns the de Rahm complex into a graded algebra. i n ∗ n With the basis {dx }i=1 for the cotangent bundle T R , the coordinate expression for a 1-form α ∈ Ω1(R3), i.e. a section of T ∗Rn, is 1 n α(x) = α1(x) dx + ··· + αn(x) dx . The basis functions for higher degree forms are generated using the wedge product in Eq. 7, which ensures anti-symmetry. In R3, for example, one has 0 3 Ω (R ) 3 f = f(x) 1 3 1 2 3 Ω (R ) 3 α = α1(x) dx + α2(x) dx + α3(x) dx 2 3 2 3 3 1 1 2 Ω (R ) 3 β = β1(x) dx ∧ dx + β2(x) dx ∧ dx + β3(x) dx ∧ dx 3 3 1 2 3 Ω (R ) 3 γ = γ(x) dx ∧ dx ∧ dx .

For an arbitrary form α ∈ Ωr(Rn) the coordinate expression is

X j1 jr α(x) = αj1,··· ,jr (x) dx ∧ · · · ∧ dx

j1<···

9 with the summation being over all ordered sequences j1 < ··· < jr, which again ensures the anti-symmetry of the form. Except for our discussions of Stokes’ theorem at the end of this section and in

Sec. 5.2.1, we will assume that the coordinate functions αj1,··· ,jr (x) are smooth, or at least sufficiently smooth for the operation under consideration, and real- valued. Differential forms with coordinate functions that are distributions in the sense of Schwartz were introduced by de Rahm [66] and are known as ‘currents’. We will denote the space of such (n − r)-forms as S0(Ωn−r, Rn) and they are dual to the space S(Ωr, Rn) of differential r-forms whose coordinate functions are test functions. The duality pairing hh·, ·ii between S0(Ωr, Rn) and S(Ωr, Rn) is, as in the scalar case, analogous to the L2-inner product for differential forms, which will be introduced in the next paragraph. Hilbert Space Structure and Hodge Dual. So far we did not require a metric and this is an important feature of the exterior calculus. When a metric is available, as is the case in Rn, the exterior calculus can be equipped with additional structure. We will introduce this structure next. 0 n Let a non-degenerate, symmetric, covariant tensor field g ∈ T2 (R ) of rank −1 2 n 2 be a given metric and g ∈ T0 (R ) be the associated co-metric acting on ij co-vectors; the components will be denoted as gij and g , respectively. Unless mentioned otherwise, we will work with the canonical metric on Rn (which, however, is not the identity in spherical coordinates). The metric g and its dual g−1 can be used to identify vectors and co-vectors by “raising” and “lowering” indices. Let X,Y ∈ X(Rn) and α, β ∈ X∗(Rn) =∼ Ω1(Rn). Then [ i i [ i 1 n X (Y ) ≡ g(X,Y ) = gijX Y ⇔ (X )j = gij X ∈ Ω (R ) ] ij ] j ij n α (β) ≡ g(α, β) = g αi βj ⇔ (α ) = g αi ∈ X(R ). The flat operator [ and the sharp operator ] introduced above are together known as musical isomorphism. With these, the classical differential operators from vector calculus can be related to the exterior derivative. Let f ∈ F(R3) and u ∈ X(R3) be a vector field. Then ∇f = (df)] (9a) ∇ × u = (?(du[))]. (9b) ∇ · u = ?d(?u[). (9c)

The metric induces, next to the musical isomorphism, also an L2-inner product hh·, ·ii for diﬀerential r-forms. Let α, β ∈ Ωr(Rn). Then Z j1,··· ,jr 1 n hhα, βii = αj1,··· ,jr β dx ∧ · · · ∧ dx (10a) n R where the contra-variant tensor components βj1,··· ,jr are obtained by index raising with the metric,

j1,··· ,jr j1,i1 jr ,ir β = g ··· g βi1,··· ,ir . (10b)

10 r n The space L2(Ω , R ) is defined in the usual way as all differential r-forms with finite L2 norm. Unless stated otherwise (e.g. for Stokes’ theorem), we will r n assume that all forms we work with are in L2(Ω , R ). Eq. 10 can equivalently be written using the Hodge dual ?β as Z hhα, βii = α ∧ ?β n R and the equality with Eq. 10 defines ?β. Since the integral in the last equation has to be over a volume form, i.e. one of maximum degree r = n, the Hodge dual has to satisfy

r n n−r n ? :Ω (R ) → Ω (R ) and it also holds that ? ? α = (−1)n(n−r)α. By linearity of the Hodge dual, which follows from those of the contraction in Eq. 10b and the wedge product, for the coordinate expression of ?β it suﬃces to know those for the form basis functions. For R3, for example, these are ?1 = dx1 ∧ dx2 ∧ dx3 (11a) ?dx1 = dx2 ∧ dx3 (11b) ?dx2 = −dx1 ∧ dx3 (11c) ?dx3 = dx1 ∧ dx2. (11d)

4.3. The Fourier Transform of the Exterior Algebra Motivation. Before introducing the formal definition of the Fourier transform of differential forms, it is instructive to understand the duality that underlies it. The classical result that integration becomes under the Fourier transform pointwise evaluation, i.e. Z f(x) dx = fˆ(0), (12) n R 1 n n n suggests that the volume form f(x) dx = f1···ndx ∧ · · · ∧ dx ∈ Ω (R ) should map to a 0-form in frequency space, since for 0-forms “integration” amounts to pointwise evaluation, cf. [65]. This is also required by covariance because if the volume form f(x) dx would map to another volume form (or to a form of degree other than zero) under the Fourier transform then it would “pick up” a Jacobian term under coordinate transformations. But fˆ(0) is already the integrated value and hence invariant under changes of coordinates (e.g. the mass of an object or the energy of a field). This observation for volume forms can be consistently extended to differential forms of lower degree. For example, for a plane in R3, which from the point of view of the exterior algebra is a 2-form, the Fourier transform is known to be in the direction of the plane’s normal, i.e. a 1-form in this direction in frequency space. For the x1-x2 plane we have, for example, F δ1,2(x) = δ3(ξ)

11 3 where δ1,2(x) is the distribution representing the x1-x2 plane in R and δ3(ξ) its Fourier transform in the ξ3 direction. With the above considerations, the duality between diﬀerential forms in space and frequency is expected to be

where we denote by Ωbn−r(Rbn) the exterior algebra in frequency space Rbn. The spatial coordinate x = xi ∂/∂xi ∈ Rn pairs with the covector ξ = i i n ix ξi ξi dx ∈ Rb in the complex exponential e of the Fourier transform. When we require that the usual correspondence between differential operators ∂/∂xi i and their Fourier transform ξidx (no summation) is to hold (in an appropriate sense, see Proposition 4 below) we also have to expect that the exterior algebra in frequency space is naturally defined over ∂/∂ξi ≡ ∂/∂xi where we use the notation ∂/∂ξi to be explicit if we are considering the primary or dual domain with respect to the Fourier transform. Differential forms in frequency space are hence defined over form basis functions ∂/∂ξj1 ∧ · · · ∧ ∂/∂ξjr and this provides the second cornerstone for the definition of the Fourier transform of differential forms. The Fourier Transform of Differential Forms. With the above motivation, we define the Fourier transform of differential forms, following an approach introduced in mathematical physics [63, 67, 64] based on the theory of super vector spaces. Definition 1. Let α ∈ Ωr(Rn) be a differential form of degree r,

X j1 jr α(x) = αj1,··· ,jr (x) dx ∧ · · · ∧ dx .

j1<···

n−r n Then the Fourier transform αb = F(α) ∈ Ωb (Rb ) of α is the (n − r)-form on frequency space Rbn given by Z 1 X j j −i xpξ i dxq ∂ F(α)(ξ) = α (x) dx 1 ∧ · · · ∧ dx r ∧ e p ∧ e ∂ξq n/2 j1,··· ,jr (2π) n j1<···

12 From a functional analytic perspective, the usual requirements for the exis- tence of the Fourier transform apply with respect to the coordinate functions α (x) and αj1,··· ,jr (ξ). j1,··· ,jr b dxq ∂ Remark 1. In the literature, e.g. [63, 64], the exponential e ∂ξq of the form basis functions is typically defined with a complex unit in the exponent, which is considered there as only serving “aesthetic” purposes to visually closer match the scalar case. For real-valued functions this is, however, not true and the use of idxq ∂ e ∂ξq leads to inconsistencies in the inverse Fourier transform. In particular, there are no differential r-form wavelets that satisfy the de Rahm complex and that are real-valued in space. We hence use the definition above without the complex unit. To evaluate the exponential in Eq. 13 one uses that the exterior algebra has a Z2-graded structure, with the even grade given by the differential forms of even degree and the odd one by differential forms of odd degree.2 The Z2-grading is respected in the tensor product by using the following multiplication rule (for homogenous elements):

par(γ)par(βb) α ⊗ βb ∧x,ξ γ ⊗ δb = (−1) α ∧x γ ⊗ βb ∧ δb , (14a) e.g. [68, Ch. III.4] or [64]. Here par(·) is the parity, which is 1 when the degree of a diﬀerential form is odd and 0 otherwise. Since the order of the spaces matters for the multiplication rule, one has for the inverse Fourier transform

par(γ)par(β) αb ⊗ β ∧ξ,x γb ⊗ δ = (−1) b αb ∧ γb ⊗ β ∧x δ . (14b) Example 1. In R2 the highest order term in the exponential of the form basis functions in Eq. 13 yields

∂ 2 ∂ ∂ ∂ ∂ dxq ⊗ = dx1 ⊗ + dx2 ⊗ ∧ dx1 ⊗ + dx2 ⊗ . ∂ξq ∂ξ1 ∂ξ2 x,ξ ∂ξ1 ∂ξ2 Since the “square” terms vanish by the anti-symmetry of the wedge product this equals

∂ 2 ∂ ∂ ∂ ∂ dxq ⊗ = dx1 ⊗ ∧ dx2 ⊗ + dx2 ⊗ ∧ dx1 ⊗ ∂ξq ∂ξ1 x,ξ ∂ξ2 ∂ξ2 x,ξ ∂ξ1

∂ ∂ = (−1)1·1 dx1 ∧ dx2 ⊗ ∧ x ∂ξ1 ∂ξ2 ∂ ∂ + (−1)1·1 dx2 ∧ dx1 ⊗ ∧ x ∂ξ2 ∂ξ1 ∂ ∂ = −2 dx1 ∧ dx2 ⊗ ∧ . x ∂ξ1 ∂ξ2

2 2 With the Z -grading the exterior algebra is a super vector space. We will not pursue this viewpoint in our work.

13 To simplify the notation, we will typically not explicitly write the tensor product as above. When the exponential of the form basis functions in Eq. 13 has been expanded, the usual rules for integration of diﬀerential forms apply for evaluating the Fourier transform. In particular, the integral vanishes unless Eq. 13 together with the the basis functions dxj1 ∧· · ·∧dxjr of α yield a volume form. This is also the reason why it suﬃces to consider the power series in Eq. 13 up to n. 1 1 3 Example 2. For α = α1(x) dx ∈ Ω (R ) we want to compute the Fourier transform Z 1 1 −ix ξp dxq ∂ F(α)(ξ) = α (x) dx ∧ e p ∧ e ∂ξq . 3/2 1 (2π) n R

dxq ∂ The exponential e ∂ξq of the form basis functions expands in R3 as dxq ∂ 1 ∂ 2 ∂ 3 ∂ e ∂ξq = 1 + dx + dx + dx ∂ξ1 ∂ξ2 ∂ξ3 ∂ ∂ ∂ ∂ ∂ ∂ − dx2 ∧ dx3 ∧ + dx1 ∧ dx3 ∧ + dx1 ∧ dx2 ∧ ∂ξ2 ∂ξ3 ∂ξ1 ∂ξ3 ∂ξ1 ∂ξ2 ∂ ∂ ∂ − dx1 ∧ dx2 ∧ dx3 . ∂ξ1 ∂ξ2 ∂ξ3

1 For α = α1(x) dx the integral of the Fourier transform becomes a volume form hence only with the second line and only the first term yields a nonzero expression by the anti-symmetry of the wedge product. Thus, 1 Z i ∂ ∂ F(α)(ξ) = α (x) e−i xi ξ dx1 ∧ −dx2 ∧ dx3 ∧ 3/2 1 2 3 (2π) n ∂ξ ∂ξ R −1 ∂ ∂ = α (ξ) ∧ (2π)3/2 b1 ∂ξ2 ∂ξ3 where αb1(ξ) is the scalar Fourier transform of the coordinate function α(x). By linearity, for the differential form part of the Fourier transform it suffices to know the transform of the form basis functions. In R2 these are ∂ ∂ F1 = − ∧ (15a) x ∂ξ1 ∂ξ2 ∂ Fdxj1 = sgn (σ) (15b) ∂ξj2 1 2 F dx ∧ dx = 1ξ (15c) with the permuation σ being σ = 1 , 2 . In 3 one has j1,j2 R ∂ ∂ ∂ F1 = − ∧ ∧ (16a) x ∂ξ1 ∂ξ2 ∂ξ3

14 ∂ ∂ Fdxj1 = − sgn (σ) ∧ (16b) ∂ξj2 ∂ξj3 ∂ Fdxj1 ∧ dxj2 = sgn(σ) (16c) ∂ξj3 1 2 3 F dx ∧ dx ∧ dx = 1ξ (16d) with the permutation being σ = 1 , 2, 3 . j1,j2,j3 Remark 2. Comparing Eq. 16 to Eq. 11 we see that the Fourier transform of the differential form basis functions has a structure analogous to those of the Hodge dual. A general description of the Fourier transform of the dxj1 ∧ · · · ∧ dxjr can hence be obtained when we borrow some notation from the general description of the Hodge star [61, Ch. 7.2]. Consider a permutation j1 ··· jr ··· σ(j) = ∈ Sn σ1 ··· σr σn−r+1 ··· σn whose first r indices match the multi-index j = (j1, ··· , jr) of the given differential form basis function. Here Sn is the group of permutations of {1, ··· , n}. One then has ∂ ∂ F dxj1 ∧ · · · ∧ dxjr = −(−1)br/2c sgn(σ) ∧ · · · ∧ . ∂ξσn−r+1 ∂ξσn A direct consequence of our definition of the Fourier transform of differential forms is the following. Proposition 3. F−1 ◦ F is the identity. Proof. This can be checked with a coordinate calculation using the above rules (or [63, Proposition 5.2]). Exterior derivative. With the Fourier transform of differential forms being defined, we can consider those of the exterior derivative. It is given in the next proposition. Proposition 4. The Fourier transform (or principal symbol) of the exterior derivative d : Ωr(Rn) → Ωr+1(Rn) is the anti-derivation bd : Ωbn−r(Rbn) → n−r−1 n Ωb (Rb ) given by the interior product iiξαb, i.e. F dα)(ξ) = bd αb (ξ) = iiξαb(ξ) 1 n where ξ is the coordinate vector ξ = ξ1 dξ +···+ξn dξ . The following diagram thus commutes d Ωr(Rn) −−−−−−−−−→ Ωr+1(Rn)     Fy yF ˆ d = iiξ Ωbn−r(Rbn) −−−−−−−−−→ Ωbn−r−1(Rbn) and the Fourier transform of the de Rahm co-chain complex is a chain complex.

15 Proof. The coordinate expression of the exterior derivative is n X X ∂αj ,··· ,j dα(x) = 1 r dxp ∧ dxj1 ∧ · · · ∧ dxjr . ∂xp p=1 j1<···

For each nonzero term there is a permutation σ(j, p) = (j1, ··· , p, ··· , jr) ∈ Sn, analogous to those in Remark 2, that brings dxp in canonical, increasing order, introducing a sign factor sgn(σ(jp)) through the anti-symmetry of the wedge product. By linearity it suﬃces to consider the pth term in the following. Its Fourier transform is

X p ∂ ∂ F dαp = s(r + 1) iξ αj1,··· ,jr sgn(σ(jp)) ∧ · · · ∧ b ∂ξσ(jp)n−r+2 ∂ξσ(jp)n j1<···

1. In the literature the interior product iiξ with the coordinate vector iξ is also known as the Koszul diﬀerential, see e.g. [49, Ch. 7.2]. 2. The position vector ξ in the symbol bd = iiξ of the exterior derivative is naturally expressed in spherical coordinates, where ξ = |ξ| dˆr and dˆr is the radial basis vector. Proposition 4 hence suggests that spherical coordinates are natural for working with the de Rahm complex in frequency space. This is a key observation that underlies the construction of our local spectral exterior calculus in the next section. It will also play an important role for the description of the Hodge-Helmholtz decomposition in frequency space in the following.

16 1 2 3 1 3 2 3 Example 3. Let α(x) = α1dx +α2dx +α3dx ∈ Ω (R ). Then dα(x) ∈ Ω (R ) and it corresponds to the curl of α] in classical vector calculus, cf. Eq. 9. In frequency space, α(x) is the 2-form ∂ ∂ ∂ ∂ ∂ ∂ α(ξ) = −α (ξ) ∧ + α (ξ) ∧ − α (ξ) ∧ , b b1 ∂ξ2 ∂ξ3 b2 ∂ξ1 ∂ξ3 b3 ∂ξ1 ∂ξ2 cf. Example 2. The interior product that deﬁnes the exterior derivative in frequency space is evaluated using the Leibniz rule for it and linearity. For example, for the ﬁrst term of the last equation this yields ∂ ∂ ∂ ∂ i − α (ξ) ∧ = −α (ξ) i ∧ iξ b1 ∂ξ2 ∂ξ3 b1 iξ ∂ξ2 ∂ξ3 ∂ ∂ ∂ ∂ = −α (ξ) i ∧ − ∧ i b1 iξ ∂ξ2 ∂ξ3 ∂ξ2 iξ ∂ξ3 ∂ ∂ = −α (ξ) iξ − iξ b1 2 ∂ξ3 3 ∂ξ2 With the analogous calculations also for the other terms we obtain ∂ ∂ −i α(ξ) = −iξ α (ξ) + iξ α (ξ) iξ b 1 b2 ∂ξ3 1 b3 ∂ξ2 ∂ ∂ + iξ α (ξ) − iξ α (ξ) 2 b1 ∂ξ3 2 b3 ∂ξ1 ∂ ∂ − iξ α (ξ) + iξ α (ξ) . 3 b1 ∂ξ2 3 b2 ∂ξ1 Using the musical isomorphism, the last equation can also be written as

[ ~ [ iξαb (ξ) = iξ × αb (ξ). This is the classical expression for the curl in the Fourier domain. Remark 4. In the literature, the principal symbol of the exterior derivative is also given as the exterior product

0 dcα (ξ) = iξ ∧ α,b (18) see e.g. [70, Vol. 1, I.7], [71, Sec. 3.2], or [61, Ch. 8.5] (the prime denoting the alternative definition). The expression is obtained when one uses the following definition of the Fourier transform of differential forms, e.g. [72, Sec. 3], ! X X α0(ξ) = F0 α dxj1 ··· dxjr = α (ξ) d0ξj1 ··· d0ξjr (19) b j1···jr bj1···jr j1···jr j1···jr where α (ξ) is the usual scalar Fourier transform of the coordinate functions. bj1···jr In contrast to Def. 1, with Eq. 19 the Fourier transform of an r-form is hence

17 again an r-form. A straight forward coordinate calculation shows that Eq. 18 holds with this definition. The expression can be related to ours in Proposition 4 using what in the literature is sometimes referred to as Hirani’s formula [50, 73], r(n−r) [ iX α = (−1) ? (?α ∧ X ), where α is an r-form and X a vector field. Hence, the interior product iiξαb is mapped to the wedge product iξ ∧ αb and by interpreting the d0ξi in Eq. 19 as contra-variant vectors both formulations are equivalent. In light of Eq. 12 and the discussion at the beginning of the section, it appears to us more natural that the Fourier transform of an r-form is an (n − r)-form and this is also what follows naturally from the definition of the exponential of the form basis functions in Def. 1 based on the existing work in mathematical physics [67, 63].

Wedge Product. In the scalar case, multiplication in space becomes convolution in the frequency domain. The next proposition, which also follows [63], shows that this carries over to the wedge product.

Proposition 5. Let α ∈ Ωr(Rn) and β ∈ Ωl(Rn) and let the convolution ∗ : Ωr × Ωl → Ωr+l−n of diﬀerential forms be Z (α ∗ β)(dy) = α(dx) ∧ β(dy − dx) n R where α(dx) denotes the basis representation of α with respect to the diﬀerentials dxi. Then the Fourier transform ∧ˆ of the wedge product ∧ is given by

ˆ n−r−l n F(α ∧ β) = αb ∧ βb = αb ∗ βb ∈ Ωb (Rb ). Conversely, the inverse Fourier transform ∧ˇ of the wedge product in frequency space is

−1 ˇ r+l−n n F (αb ∧ βb) = α ∧ β = α ∗ β ∈ Ω (R ). Hence the following diagram holds

r+l−n r l ∧ r+l Ω ∗ Ω × Ω Ω

F F F

∧ˆ = ∗ Ωb2n−r−l Ωbn−r × Ωbn−l Ωbn−r−l ∧ = ∗ˆ Proof. This can be proved by a direct calculation. See again [63]. Remark 5. It follows from the above definition and the usual properties of the wedge product that the convolution vanishes unless r+l ≥ n. For r+l < n one is interested in the convolution over a sub-manifold and working with differential forms requires one to make this explicit (i.e. to first pullback the form to the sub-manifold and then perform the integration there).

18 1 2 2 3 1 2 3 Example 4. Let α = α3 dx ∧ dx ∈ Ω (R ) and β = β1 dx + β2 dx + β3 dx ∈ Ω1(R3). Then Z (α ∗ β)(dy) = α(dx) ∧ β(dy − dx) n R P3 i i with β(dy − dx) = i=1 βi(x − y)(dy − dx ). Using linearity and the anti- symmetry of the wedge product we obtain Z 1 2 3 3 (α ∗ β)(dy) = α3(x) dx ∧ dx ∧ β3(x − y) (dy − dx ) n R Z 1 2 3 = α3(x) β3(x − y) dx ∧ dx ∧ dx n R

= α3 ∗ β3 where the convolution in the last line is those of scalar functions.

Hodge star and codiﬀerential. The following result characterizes the Fourier transform of the Hodge dual.

Proposition 6. Let α ∈ Ωr(Rn). Then

br/2c X ∂ ∂ F ? α = ?ˆ α = −(−1) αˆj ,··· ,j sn(σ) ? ∧ · · · ∧ b 1 r ∂ξσn−r+1 ∂ξσn j1<···Hodge star operator together with Proposition 4 implies that the Fourier transform of the codiﬀerential of a form α ∈ Ωr(Rn) is

n−1 δαc = (−1) ? iiξ ? α.b Plancherel theorem. The Plancherel theorem is a central result for the scalar Fourier transform. The next proposition establishes the analogue for diﬀerential forms.

Proposition 7 (Plancherel theorem for diﬀerential forms). With α, β ∈ Ωr(Rn),

α, β = α,b βb .

19 Proof. As in the classical case, the results follows from a direct calculation using Z −1 −1 −1 −1 α, β = F (α) , F (βb) = F (α) ∧ ? F (βb). b n b R 3 3 We will show it for α = α3(x) dx , β = β3(x) dx . The required Fourier transforms are in this case ∂ ∂ ∂ ∂ α = −α3(x) ∧ ?βc = βb3(x) ? βb = −βb3(x) . b b ∂ξ1 ∂ξ2 ∂ξ3 ∂ξ3 Then Z 3 1 2 α , β = α3(x) dx ∧ β3(x) dx ∧ dx 3 R Z 1 Z ∂ ∂ ∂ = α (ξ) eihξ,xi ∧ ∧ dx3 3/2 b3 1 2 3 3 (2π) 3 ∂ξ ∂ξ ∂ξ R Rb −1 Z ∂ ∂ ∂ ∧ β∗(η) e−ihη,xi ∧ − ∧ dx1 ∧ dx2 . x 3/2 b3 3 1 2 x (2π) 3 ∂η ∂η ∂η Rb With the graded, anti-symmetric multiplication we have Z 1 Z ∂ ∂ ∂ α , β = α (ξ) eihξ,xi ∧ ∧ ⊗ dx3 3/2 b3 1 2 3 3 (2π) 3 ∂ξ ∂ξ ∂ξ R Rb 1 Z ∂ ∂ ∂ ∧ β∗(η) e−ihη,xi ∧ ∧ ⊗ dx1 ∧ dx2 . x 3/2 b3 1 2 3 x (2π) 3 ∂η ∂η ∂η Rb Using linearity and separating the terms with x-dependence we obtain Z Z Z −1 −ihη,xi 3 ihξ,xi 1 2 α , β = 3 e dx ∧x e dx ∧x dx (2π) 3 3 3 Rb Rb R ∂ ∂ ∂ ∗ ∂ ∂ ∂ × α3(ξ) ∧ ∧ βb (η) ∧ ∧ . b ∂ξ1 ∂ξ2 ∂ξ3 3 ∂η1 ∂η2 ∂η3 Evaluating the integral over x yields Z Z ∂ ∂ ∂ α , β = − δ(ξ − η) α3(ξ) 1 ∧ 2 ∧ 3 3 3 b ∂ξ ∂ξ ∂ξ Rb Rb ∂ ∂ ∂ × βb∗(η) ∧ ∧ 3 ∂η1 ∂η2 ∂η3 Finally, with the integration over η we have Z ∗ ∂ ∂ ∂ α , β = − α3(ξ) iβb3 (ξ) 1 ∧ 2 ∧ 3 3 b ∂ξ ∂ξ ∂ξ Rb Z ∂ ∂ ∗ ∂ = α3(ξ) 1 ∧ 2 ∧ −βb3 (ξ) 3 3 b ∂ξ ∂ξ ∂ξ Rb | {z } | {z } αb ?βb∗

= αb , βb . The other cases follow by analogous calculations.

Hodge-Helmholtz decomposition. The Hodge-Helmholtz decomposition provides a characterization of the domain and image of the exterior derivative. It will be central to our formulation of the local spectral exterior calculus.

r n Theorem 1 (Hodge-Helmholtz decomposition). Let L2(Ω , R ) be the space of n diﬀerential forms of degree r on R with ﬁnite L2-norm. Then

r n r n r n L2(Ω , R ) = L2(Ωd, R ) ⊕ L2(Ωδ, R ) (20)

r n r n where L2(Ωd, R ) is the L2-space of exact forms and L2(Ωδ, R ) those of co- r n exact forms. Hence any ω ∈ L2(Ω , R ) can be written as ω = ωd +ωδ = dα+δβ r−1 n r+1 n where α ∈ L2(Ωd , R ) and β ∈ L2(Ωδ , R ). In frequency space,

n−r n n−r n n−r n L2(Ωb , Rb ) = L2(Ωbd , Rb ) ⊕ L2(Ωbδ , Rb ), (21) i.e. ωb = ωbd + ωbδ with X ∂ ∂ ω = (ω ) ∧ · · · ∧ (22a) bd bd j1,··· ,jn−r ∂θˆj1 ∂θˆjn−r j1<···

21 Homogeneous Sobolev spaces for differential forms. With the Hodge-Helmholtz decomposition and the Plancherel theorem we can introduce homogeneous Sobolev spaces for differential forms. These will provide the functional analytic setting of our local spectral exterior calculus. As we will clarify shortly, it is natural to restrict the definition to co-exact forms.

˙ 1 r n Deﬁnition 2. The homogeneous Sobolev space L2(Ωδ, R ) is n o ˙ 1 r n r n 2 L2(Ωδ, R ) = α ∈ Ωδ(R ) kαk ˙ 1 = α , α ˙ 1 < ∞ . (23a) L2 L2

˙ 1 r n where the L2(Ωδ, R )-inner product is given by Z 2 ∗ α , β L˙ 1 = dα , dβ L = |ξ| α ∧ ?βb . (23b) 2 2 n b Rb The frequency definition on the right hand side of Eq. 23b, where no interior ˆ products d = iiξ appears, can be established using a straightforward coordinate calculation. ˙ 1 n In the scalar case L2(R ), the definition of homogeneous Sobolev spaces typically uses co-sets [f]0 of functions modulo constants, i.e. 0 order polynomials, since this is required for the spaces to be Hilbert, see Appendix A.2. For a compact domain U ⊂ Rn, the Hodge-Helmholtz decomposition for functions is 0 0 Ω (U) = H(U) ⊕ Ωδ(U), where H(U) are the harmonic ones satisfying ∆f = 0, and for a star-shaped domain this are exactly the constants. The classical definition hence works with the co-exact 0-forms modulo harmonic ones. In Def. 2 we work directly with co-exact differentials, which, as the following result shows, ˙ 1 n r also ensures that L2(R , Ω ) is Hilbert. ˙ 1 r n Proposition 8. For n = 2, 3, the homogeneous Sobolev space L2(Ωδ, R ) is a Hilbert space.

Proof. With the deﬁnition of the Hodge dual in terms of the metric tensor [61, Ch. 7.2], Eq. 23b can be written as

Z n Z 2 ∗ X 2 j n |ξ| α ∧ ?βb = |ξ| αj(ξ) βb (ξ) dξ n b n b Rb j=1 Rb where dξn is the canonical volume form on Rn. But thus the scalar result [75, Ch. II.6] applies to each coordinate function. Def. 2 is only for co-exact diﬀerential forms. It can be combined with the r n usual L2-space for exact ones, i.e. L2(Ωd, R ), and in light of the de Rahm complex where exact forms are the image of co-exact ones this is the natural com- ˙ 1 r n ˙ 0 r n ∼ r n plement to L2(Ωδ, R ). For conciseness we will write L2(Ωd, R ) = L2(Ωd, R ) r n ˙ 1 r n in the following. Working with L2(Ωd, R ) and L2(Ωδ, R ) is also compatible with the classical spaces H(curl, Rn) and H(div, Rn), as the following remark shows.

22 n n Remark 6. In numerical analysis, the spaces H(curl, R ) and H(div, R ) of L2- vector fields whose curl respectively divergence are also in L2 provide typically the functional analytic setting for the discretization of vector-valued partial differential equations, e.g. [47, 76], and these are also used in finite element- based discretizations of exterior calculus [4, 49]. With the musical isomorphism to relate vector fields and 1-forms and using the Hodge-Helmholtz decomposition, H(curl, Rn) can be written as n o n n [ 1 n [ ˙ 1 1 n H(curl, R ) = u ∈ X(R ) ud ∈ L2(Ωd, R ), uδ ∈ L2(Ωδ, R )

[ [ where no control of the derivative of the exact part ud is necessary since dud = 0 and the L2-condition for the co-exact one is automatically satisﬁed. Thus, n 1 n ] ˙ 1 1 n ] H(curl, R ) = L2(Ωd, R ) ⊕ L2(Ωδ, R ) with the musical isomorphism on the right hand side understood element-wise. Analogously we have (in R2 the divergence corresponds to d ? u[) 2 ˙ 1 1 2 ] 1 2 ] H(div, R ) = L2(?Ωd, R ) ⊕ L2(Ωδ, R ) 3 2 3 ] ˙ 1 2 3 ] H(div, R ) = L2(Ωd, R ) ⊕ L2(Ωδ, R ) . We will return to the last equations in the next section. Def. 2 can be generalized to homogeneous Sobolev spaces of arbitrary order ˙ 1 r n s ∈ R, cf. [72], but L2(Ωδ, R ) suﬃces for our purposes and we hence leave this to future work.

Stokes’ theorem. An important result in the exterior calculus is Stokes’ theorem, which, up to the musical isomorphism, includes the classical theorems of Green, Gauss, and Stokes in vector calculus as special cases. Let M be an r-dimensional, smooth, compact sub-manifold of Rn with smooth boundary ∂M and let α ∈ Ωr−1(Rn). Then Stokes’ theorem states that [61, Theorem 8.2.8] Z Z i∗α = dα (24) ∂M M where i : ∂M → M is the inclusion map (cf. [65, Ch. 2.7]). The above formulation of Stokes’ theorem is not well suited for the Fourier theory of diﬀerential forms introduced in the foregoing. We hence rewrite it in a more amenable one using the characteristic functions χM and χ∂M of M and the boundary ∂M, respectively. Since the manifold M is smooth, we can always choose local coordinatesx ˜i such that M is locally aligned with the ﬁrst n r coordinatesx ˜1, ··· , x˜r of R . This carries over to dα, which has then the representation α =α ˜(˜x) d˜x1 ∧ · · · ∧ d˜xr for some coordinate functionα ˜(˜x). The form can be extended along the normal bundle of M to a smooth, compactly supported test form dα ∈ S(Ωr, Rn) in the Schwartz space S(Ωr, Rn) supported

23 in a local neighborhood of M in the ambient space Rn (e.g. [77, Lemma 2.27]); by abuse of notation we will not distinguish dα and its smooth extension. This r-form dα can then be completed to a volume form (that can be integrated) by 0 n−r n ∧-multiplying it with the characteristic differential form χM ∈ S (Ω , R ) of M, i.e. a distribution-valued (n − r)-form whose support is M. In coordinates, it is given by the partial contraction ∂ ∂ χ = dx1 ∧ · · · ∧ dxn , ··· , , ··· , (25) M ∂u1 ∂ur where the ∂/∂ui ∈ X(Rn) span the tangent space T M and have support only on M , which reflects the distributional character of χ , and dx1 ∧ · · · ∧ dxn is M the canonical volume form on Rn. Example 5. Let ∂M = S1 ⊂ R2 be the unit circle and α ∈ Ω1(S1) with polar coordinate representation α = α(θ, φ) dθ. Then the 1-form in Ω1(R2) obtained by a smooth extension in the normal direction is α =α ˜(θ, φ) dθ ∈ Ω1(R2). The 1 2 2 tangent vector field ~u of S embedded into R is ~u = δS1 (x) ∂/∂θ ∈ X(R ) where 0 n the Dirac distribution δS1 (x) ∈ S (R ) is defined by Z hδS1 , ϕi = ϕ (26) S1 for all test functions ϕ ∈ S(R2), cf. [78]. The characteristic 1-form completing α is by Eq. 25 thus ∂ χ = dθ ∧ dr δS1 (x) = δS1 (x) dr. S1 ∂θ The integral of α over S1 can hence also be written as Z Z Z α = α˜(x) dθ ∧ δS1 (x) dr = α˜(x) δS1 (x) dθ ∧ dr. 1 2 2 S R R With the above construction, the right hand side of Stokes’ theorem in Eq. 24 becomes Z Z

dα = dα ∧ χM n M R where on the right we have a weak pairing between S(Ωr, Rn) and S0(Ωn−r, Rn). To apply the formulation also to the left hand side of Stokes’ theorem we use the change of variables theorem [61, Theorem 8.1.7], Z Z i∗α = α ∂M i(∂M) where i(∂M) is the sub-manifold in Rn spanned by the boundary ∂M. The argument for M then carries over since both M and ∂M are sub-manifolds of Rn. With this, we obtain the following form for Stokes’ theorem Z Z

α ∧ χ∂M = dα ∧ χM . (27) n n R R that provides a weak formulation of the classical one in Eq. 24.

24 0 n−r n Example 6. Let r = n. The characteristic form χM ∈ S (Ω , R ) completing dα ∈ S(Ωr, Rn) to a volume form is then a 0-form, i.e. the usual characteristic function of M. In this case it is well known from the classical divergence theorem that χ∂M is given by the weak gradient of χM , cf. [79, p. 73], i.e.

i χ∂M = δ∂M(x) ni d˜x

where the ni are the components of the normal of ∂M and δ∂M (x) is the Dirac distribution deﬁned analogously to Eq. 26. In other words, when r = n then

χ∂M is nothing but the wavefront set WF(χM ) of χM [80, Ch. VI]. We will return to the connection with the wavefront set in Sec. 5.

The form of Stokes’ theorem in Eq. 27 can also be written using the L2- pairing for diﬀerential forms, i.e. using the Hodge duals of χ∂M and χM . It then takes the form

α , χT ∂M = dα , χT M . (28)

It follows from Eq. 25 that ?χ∂M = χT ∂M is a local representation for the tangent 3 space of ∂M and similarly ?χM = χT M . In R , for example, let the local tangent 3 space of a 2-surface be aligned with the x1-x2 plane. Then χ∂M = δ(x3) dx 1 2 and χT ∂M = δ(x3) dx ∧ dx . Comparing the standard form of Stokes’ theorem in Eq. 24 with the one in Eq. 28 one sees that both provide conceptually a very similar geometric description: the left hand side is the pairing of α with the tangent space T ∂M of the boundary and the right hand side those of dα with T M. However, in contrast to Eq. 24, Eq. 27 can be transferred to a frequency description using the Plancherel theorem. This yields Z Z ˆ α ∧ χ∂M = dα ∧ χM . (29) n b b n b b Rb Rb To obtain some insight into this formulation of Stokes’ theorem, we will consider a concrete example. 2 1 1 2 Example 7. Let M = B , ∂M = S , and α = αθ(r) dθ ∈ Ω (B ). Thus dα = −(∂α/∂r) dθ ∧ dr. From Example 5 we already know that the completion of α to a volume form is given by χ = χ = δ 1 (r) dr. The completion of ∂M S1 S dα is the 0-form χ = χ (r). Stokes’ theorem in the form in Eq. 27 then M B2 becomes Z Z ∂αθ(r) α (r) dθ ∧ δ 1 (r) dr = − dθ ∧ dr ∧ χ (r) θ S B2 2 2 ∂r R R and re-arranging terms yields Z Z ∂αθ(r) α (r) δ 1 (r) dθ ∧ dr = − χ (r) dθ ∧ dr. θ S B2 2 2 ∂r R R The equality can now easily be deduced using integration by parts.

25 To write the last equations in the frequency domain as in Eq. 29, the following Fourier transforms are required ∂ α = α(ˆr) b b ∂rˆ ∂ ∂ χ = δb 1 (ξ) = −J1(ˆr) b∂M S ∂θˆ ∂θˆ d∂α dcα = (ˆr) = α(ˆr) irˆ ∂r b

∂ ∂ J1(ˆr) ∂ ∂ χ = δb 2 (ξ) ∧ = ∧ bM B ∂θˆ ∂rˆ rˆ ∂θˆ ∂rˆ ˆ where (θ, rˆ) are polar coordinates in frequency space, J1(·) is the (cylindrical) Bessel function of order 1, and for dcα we used Proposition 4 and that ξ =r ˆdˆr in polar coordinates. Inserting the Fourier transforms into Eq. 29 we obtain Z Z ∂ ∂ J1(ˆr) ∂ ∂ − α(ˆr) ∧ J1(ˆr) = α(ˆr)r ˆ ∧ ∧ . 2 b ∂rˆ ˆ 2 b rˆ ˆ ∂rˆ Rb ∂θ Rb ∂θ Rearranging the terms yields Z Z ∂ ∂ J1(ˆr) ∂ ∂ α(ˆr) J1(ˆr) ∧ = α(ˆr)r ˆ ∧ 2 b ˆ ∂rˆ 2 b rˆ ˆ ∂rˆ Rb ∂θ Rb ∂θ and cancelling ther ˆ factor on the right hand side shows that the equality holds. The above example provides insight into the mechanics behind Stokes’ theorem in the Fourier domain: The linear growth by |ξ| in dcα compared to αb, which is introduced by the exterior derivative, is compensated by the diﬀerence in the decay rates of χ and χ . One hence has an interesting interweaving b∂M bM of functional analytic and geometric aspects in Eq. 27 and Eq. 29 that is not apparent in the classical form of Stokes’ theorem.

5. Ψec(Rn): A Local Spectral Exterior Calculus In the last section, we studied the Fourier transform of differential forms and saw that the symbol of the exterior derivative is a radial operator, i.e. it acts on the radial part of a differential form’s Fourier transform. This structure aligned with spherical coordinates in the frequency domain is also reflected in the Hodge-Helmholtz decomposition, with an exact differential form being one those Fourier transform has no radial component. Motivated by this, we will define the polar differential form wavelets of Ψec(Rn) in spherical coordinates in Fourier space. Compatible window functions providing the wavelet’s space- frequency localization are given by the polar wavelets of Sec. 3 and these ensure the admissibility of the frames, allow for flexible angular localization, and yield differential form wavelets that have closed form expressions in the spatial domain. The functional analytic setting for our construction is motivated by the

26 2 2 2

- - - 2 2 2 2 2 2

- - - 2 2 2

2 2 2

- - - 2 2 2 2 2 2

- - - 2 2 2

1,ν 1,ν,2 Figure 1: Examples for differential 1-form mother wavelets ψj ≡ ψj,0,0 for different levels 1,d,2 1,δ,2 with exact forms ψj,0,0 in the top row and co-exact ones ψj,0,0 in the bottom row (visualized as vector fields). The anisotropic functions exist at different directions and model discontinuities aligned with their orientation (here along the x1-axis), cf. Sec. 5.2.1 and Example 9.

27 2 Figure 2: Structure of Ψec(R ). classical spaces H(curl, Rn) and H(div, Rn) but we will from the outset distinguish between exact and co-exact differential forms, since this provides us with control over the kernel and image of the exterior derivative and the codifferen- tial, cf. Remark 6. We will hence work with the homogeneous Sobolev spaces ˙ |ν| r n L2 (Ων , R ) introduced in Def. 2. In the following, we will formally define differential forms wavelet and study their properties. Stokes’ theorem and other results related to differential forms are considered in the sequel. To not unnecessarily clutter the presentation, we collect proofs in Sec. 5.4.

5.1. Polar Differential Form Wavelets We begin by defining polar differential form wavelets. Definition 3. Let {∂/∂θˆ, ∂/∂rˆ} and {∂/∂θˆ, ∂/∂φˆ, ∂/∂rˆ} be orthonormal frames in spherical coordinates for T ∗Rb2 and T ∗Rb3, respectively, with the first ∗ n−1 n − 1 coordinates spanning T Sξ . Furthermore, let ν ∈ {d, δ}, with |d| = 0 and |δ| = 1, and the index a satisfy a ∈ An,r,ν with

{1, 2} n = 3 and r = 1, ν = δ; r = 2, ν = d A = n,r,ν {1} otherwise

ˆ When ψs(ξ) is a polar wavelet in the sense of Sec. 3 then the polar diﬀerential r- r,ν,n n forms wavelets ψs,a (x) in R , deﬁned in frequency space through their Fourier ˆr,ν,n transform ψs,a (ξ), are as given in Fig. 2 and Fig. 3.

Remark 7. Remarks on Def. 3: 1. The tangent space to the sphere Sˆ2 in frequency space is 2-dimensional and hence there are two diﬀerent wavelets that have a tangential component

28 3 Figure 3: Structure of Ψec(R ).

2 to Sˆ . Ar,ν is used to index these. To simplify notation we will typically omit a when a ∈ {1} and the same holds when n is clear from the context. 2. As discussed in Sec. 4.3, the Fourier transform of an r-form is an (n − r)- form in frequency space. Although they are defined in the Fourier domain, ˆr,ν,n we use the degree of the form in space in our nomenclature, i.e. ψs,a ∈ Ωbn−r(Rbn). 3. The above definition can be extended to differential forms on R1. There ˆ0,δ,1 one only has a radial direction and the wavelets are hence given by ψs = ˆ ˆ1,d,1 ˆ −iψs(ξ) ∂/∂ξ and ψs = ψs(ξ). This extension is useful, for example, when one considers fiber integration, cf. Sec. 5.2.2. It follows immediately from the definition that differential r-forms wavelets are differential forms in sense of the continuous theory. This is a vital difference to existing discretizations of exterior calculus, e.g. [9, 49]. We will return to this point in the sequel. The principal utility of differential form wavelets stems from the following result.

n Theorem 2. Let {ψs}s∈S be a scalar, tight polar wavelet frame for L2(R ), i.e. Proposition 1 or Proposition 2 holds. Then the induced polar diﬀerential r,ν,n ˙ |ν| r n r-forms wavelets ψs,a provide a tight frame for L2 (Ων , R ), i.e. any r-form

29 ˙ |ν| r n α ∈ L2 (Ων , R ) has the representation

X X DD r,ν,nEE r,ν,n α(x) = α , ψs,a ψs,a (x).

a∈Ar,ν s∈S

The polar diﬀerential r-form wavelets are, furthermore, real-valued in space and have the following explicit expressions. In R2 they are: 2j X ψ0,δ(x) = imβs eimθx h (|2js x − k |) s 2π m m s m

2j X X ψ1,d(x) = imσ β eimσ θ h (|2js x − k |) −σ dx1 + i dx2) s 4π m mσ s σ∈{−1,1} m

2j X X ψ2,δ(x) = − imσ β eimσ θ h (|2js x − k |)i dx1 + σ dx2 s 4π m mσ s σ∈{−1,1} m

2j X ψ2,d(x) = imβs eimθx h (|2js x − k |) dx1 ∧ dx2 s 2π m m s m ˆ where mσ = m + σ and hm(·) is the Hankel transform of h(|ξ|) of order m. In R3 the diﬀerential r-form wavelets are:

−23j/2 X ψ0,δ(x) = il κs y (¯x) h1(|2js x − k |) s 8π2 lm lm l s l,m

3j/2 3 −2 X X j2,j3 |ν| ψ1,ν (x) = sgn(σ ) il ψˆ1,ν y (¯x) h (|2js x − k |) dxj1 s,a 8π2 j s,a lm lm l s j1=1 l,m

3j/2 3 −2 X X j1 |ν| ψ2,ν (x) = sgn(σ ) ilψˆ2,ν y (¯x) h (|2js x − k |) dxj2 ∧ dxj3 s,a 8π2 j s,a lm lm l s j1=1 l,m

−23j/2 X ψ3,d(x) = il κs y (¯x) h (|2js x − k |) dx1 ∧ dx2 ∧ dx3 s 8π2 lm lm l s l,m where

j2,j3 X X ψˆ1,δ = − κs φˆj1 ·rˆj2 ) Glm s,2 lm l1m2 l2m2 l1m1;l2m2 l1,m1 l2,m2

j1 X X ψˆ2,d = κs θˆj1 Glm s,1 lm l1m2 l2m2 l1m1;l2m2 l1,m1 l2,m2

1,ν,3 1,ν,3 ˆj ˆj j and analogous for the other cases of ψs,a and ψs,a . The φ , φ , rˆ are the components of the spherical form basis functions ∂/∂θˆ, ∂/∂φˆ, ∂/∂rˆ with respect

30 j q to the Cartesian ones ∂/∂ξ . The radial window hl (·) is given by the modified spherical Hankel transform Z (q) 1 ˆ 2 hl (r) = q h(ˆr) jl(ˆr r)r ˆ dˆr. + (ˆr) Rb r The proof is relegated to Sec. 5.4. For exact forms in Ωd, i.e. |ν| = 0, we r,ν,n r n ∼ ˙ 0 r n thus have that the ψs,a form a tight frame for L2(Ωd, R ) = L2(Ωd, R ) and r for co-exact ones in Ωδ, i.e. |ν| = 1, they are a tight frame for the homogeneous ˙ 1 r n Sobolev space L2(Ωδ, R ), see Fig. 2 and Fig. 3. The theorem ensures that r,ν,n the polar differential form wavelets ψs,a can represent the differential forms of 2 3 r,ν,n interest in applications in R and R . Plots of some ψs,a can be found Fig. 1 q and Fig. 4. As in the scalar case, the radial window hl (·) has a closed form expression, for example, when one uses in the Fourier domain the window hˆ(|ξ|) that has been proposed for the steerable pyramid [60]. Remark 8. There are alternatives to the function spaces we used in Theorem 2. ˙ n/2 0 n For example, starting with L2 (Ωδ, R ) for functions and losing one degree ˙ n/2−r r n of regularity with every exterior derivative, i.e. working with L2 (Ων , R ), one obtains a more symmetric construction. When one then also works with ˙ −n/2+r r n the dual spaces L2 (Ωδ, R ), one has at least partial closure under the Hodge dual. Our choice is motivated by the classical spaces H(curl, Rn) and H(div, Rn), cf. Remark 6, but it is an interesting direction for future work to more systematically explore alternatives. r,ν,n The following proposition establishes that the ψs,a satisfy important properties of Cartan’s exterior calculus and hence constitute a discretization of it.

r,ν,n ˙ |ν| r n Theorem 3. The polar diﬀerential r-forms wavelet ψs,a ∈ L2 (Ων , R ) satisfy: i) Exterior derivative:

r,d,n dψs,a (x) = 0

r,δ,n r+1,d,n dψs,a (x) = ψs,a (x)

and furthermore

ψr,δ,n = ψr+1,d,n s,a ˙ 1 s,a L2 L2 ii) Codiﬀerential:

r,δ,n δψs,a (x) = 0

iii) Hodge dual:

ˆr,ν,2 2|ν|−1 ˆn−r,ν,¯ 2 ? ψs (ξ) = sgn(σ) |ξ| ψs (ξ)

31 2,d,3 3 Figure 4: Exact diﬀerential form wavelets ψs,2 in R in the frequency domain (top) and the spatial one (bottom) visualized as vector ﬁelds. The left function is isotropic around the x3-axis and the others are directional modeling high frequency features aligned with the x1-x2 plane.

ˆr,ν,3 b(r+1)/2c 2|ν|−1 ˆn−r,ν,¯ 3 ? ψs,a (ξ) = (−1) |ξ| ψs,a¯ (ξ) where σ is the permutation associated with the form basis functions of the wavelet and δ¯ = d and d¯ = δ. iv) Rigid body transformations:

J J X X r,ν,n X X r,ν,n α = αs,a ψs,a ⇔ Aα = α˜s,a ψs,a ,

a js,ks,ts a js,ks,ts

˙ |ν| r n where A ∈ E(n) is a rigid body transformation and α ∈ L2 (Ων , R ), i.e. the translation and/or rotation of a J-bandlimited representation in polar differential form wavelets can be represented in the same frame with the same bandlimit for some coefficients α˜s, The proof of the theorem is again relegated to Sec. 4.3. The theorem shows that polar differential form wavelets are closed under the exterior derivative and r,ν,n in the sense of the continuous theory since the ψs,a are continuous differential forms. They furthermore respect the Hodge-Helmholtz decomposition, since one has separate representations for exact and co-exact forms. By linearity these r,ν,n properties carry over to arbitrary forms represented in the wavelets ψs,a . This makes them useful for numerical calculations.

32 Remark 9. Remarks on Theoren 3: i) For the evaluation of the exterior derivative in the Fourier domain, i.e. ˆ d = iiξ the Leibniz rule has to be used and this leads to additional sign changes. ii) The imaginary unit in the definition of the differential 1-form wavelets ˆ1,ν,2 ψs in the Fourier domain in Fig. 2, which appears a priori because of the exterior derivative, is also required for the wavelets to be real-valued in the spatial domain, cf. the proof of the theorem. iii) For the Hodge dual one does not have closure. This arises from the different functions spaces that are used for exact and co-exact forms, which, however, are required for the closure of the exterior derivative. One remedy would be to use an alternative functional analytic setting and also work with the dual spaces, cf. Remark 8.

iv) For isotropic basis functions, i.e. when βm = δm0, the 1-form wavelets in R2 are of the form of an ideal source and an ideal divergence free vortex, i.e. for the mother wavelets one has

1,d 1,δ ψ = h1(|x|) dr ψ = h1(|x|) dθ,

see left most column of Fig. 1. where dθ and dr are the canonical form basis functions in polar coordinates in space. Analogously, in R3 one obtains in the isotropic case an axis-aligned source and vortex tube , cf. Fig. 4. v) The form basis vector ∂/∂θˆ and ∂/∂φˆ have singularities at the poles. In [45] it has been proposed to use a tight, redundant frame to span TS2 and a simple construction of such a frame was proposed there. While possible, some care is required in our context because in exterior calculus it is usually assumed that the tangent space is spanned by a (biorthogonal) basis. We leave it to future work to either re-write the necessary parts of exterior calculus using a redundant frame or ﬁnd suitable coordinate charts for S2 that work well with polar wavelets and avoid the singularities. With compactly supported directional localization windowsγ ˆ(ξ¯) one can choose {∂/∂θ,ˆ ∂/∂φˆ} such that the singularities are outside the support.

5.2. Exterior Calculus using Diﬀerential Forms Wavelets We continue with a discussion of the diﬀerential form wavelet interpretation of important results in the exterior calculus.

5.2.1. Stokes theorem We showed in Sec. 4.3 that Stokes’ theorem can be written as Z Z

α ∧ χ∂M = dα ∧ χM . n n R R 0 n−r+1 n 0 n−r n where χ∂M ∈ S (Ω , R ) and χM ∈ S (Ω , R ) are the characteristic diﬀerential forms of ∂M and M, respectively. The diﬀerential form wavelet

33 representation of α ∈ S(Ωr−1, Rn) and dα ∈ S(Ωr, Rn), which are smooth, compactly supported extensions of the original forms on ∂M and M along the normal, are given by

X r−1,δ X r,d α = αs ψs dα = αs ψs s s where we immediately used that d2 = 0 and ∂∂M = ∅ so that it suﬃces to consider the co-exact part of α. Note also that the coeﬃcients αs on both sides of the equation are identical. With the above representations and using linearity, Stokes’ theorem can be written as Z Z X r−1,δ X r,d αs ψs ∧ χ∂M = αs ψs ∧ χM . (30) n n s R s R

By interpreting the integrals as the projection of χ∂M and χM onto the wavelets we obtain

X ∂M X M αs χ˜s = αs χ˜s (31) s s

∂M M where we use the tilde to distinguishχ ˜s andχ ˜s from the frame coefficients obtained with the L2-inner product for differential forms with the Hodge dual. ∂M ∂M It follows immediately from Eq. 31 that the coefficientsχ ˜s andχ ˜s are in ∂M M fact equal, i.e.χ ˜s =χ ˜s . r,d When ψs is a volume form, i.e. r = n, then the integral on the right hand side of Eq. 30 describes the frame coefficient of the scalar characteristic function n,d χM with the volume form wavelet ψs = ψs(x) dx, cf. Theorem 3, v.), i.e we can write Z M r,d χ˜s = ψs χM dx. n R

M The behavior of these coefficientsχ ˜s has been studied extensively in the literature on ridgelets (e.g. [81, 82]), curvelets (e.g. [30, 83, 21, 22, 84]), shearlets (e.g. [85, 86, 87]), contourlets (e.g. [24, 88]) and related constructions such as α- molecules (e.g. [57, 89]). From these results it is known that, for sufficiently fine M r,d levels js, the coefficientsχ ˜s are non-negligible only when ψs is in the neighborhood of the boundary ∂M and, in the anisotropic, curvelet-like case, when M ∂M it is oriented along ∂M. By Eq. 31, the “geometry” coefficientsχ ˜s =χ ˜s thus select the “signal” coefficients αs (of α and hence also dα) in the vicinity of ∂M M so that αs will provide a significant contribution to the term only whenχ ˜s is non-negligible. This selection implements the pullback along the inclusion map i : ∂M → M in the original form of Stokes’ theorem in Eq. 24. The above argument also shows that anisotropic wavelets are more efficient for realizing M the boundary integral numerically than isotropic ones since the coefficientsχ ˜s are then sparser and the sums in Eq. 31 contain fewer non-negligible terms. In particular, when ∂M is C2 then curvelets, shearlets, and contourlets yield quasi-optimally sparse representations of χM .

34 i We already discussed in Example 5 that χ∂M = δ∂M(x) ni d˜x is just the wavefront set WF(χM ) of χM . The integral on the left hand side of Eq. 30 is r−1,δ thus a projection of WF(χM ) onto the ψs . That curvelet-like wavelets can represent, or resolve, the wavefront set has been established previously in the literature [21, 90]. The formulation of Stokes’ theorem using differential form wavelets provides, in our opinion, an interesting perspective on this work [21, 90]. Eq. 31 is valid for any r and not just the case r = n for which the existing results for curvelets, shearlets and related constructions apply. To our knowledge, there are no results on the approximation power of “vector-valued” wavelets or curvelets. We conjecture that also in the general case one has non-negligible ∂M M coefficientsχ ˜s andχ ˜s only in the vicinity of ∂M and with an orientation aligned with it. We leave a thorough investigation of this question to future work. Example 8. We consider Kelvin’s circulation theorem in fluid mechanics, e.g. [91, Ch. 1.2] or [92, Ch. 1], that relates the fluid velocity and vorticity using Stokes’ theorem. With differential forms it can be stated as Z Z u[ = ζ ∂Σ Σ where u[ is the 1-form associated with the fluid velocity field ~u ∈ X(U), ζ = du[ ∈ Ω2(U) is the vorticity, and Σ is an area embedded in the flow domain U ⊂ R2. Representing u[ using the 1-form frame,

[ X [ 1,δ u = us ψs , s the representation for the associated vorticity is

[ X [ 2,d ζ = du = us ψs s

[ and both hence have the same expansion coeﬃcients us. With Eq. 31, Kelvin’s circulation theorem thus becomes Z Z X [ 1,δ X [ 2,d us ψs ∧ χ∂Σ = us ψs ∧ χΣ . (32) 2 2 s R s R As in Example 5, the characteristic function associated with ∂Σ is the weak gradient

i χ∂Σ = dχΣ = δ∂Σ(x) ni dx where the ni are the components of the normal of ∂Σ and δ∂Σ(x) is its Dirac- distribution. In components we hence have for the integrand on the left hand side of Eq. 32,

1,δ 1,δ 1 1,δ 2 1 2 ψs ∧ χ∂Σ = ψs dx + ψs dx ∧ δ∂Σ(x) n1 dx + δ∂Σ(x) n2 dx

35 frequency4 spatial 2

3 -1 2 ζ0 4 1 x 2 4 6 8 10 1

-1

- -4 2 Figure 5: Conceptual depiction of fiber integration of differential form wavelets for a co-exact 1,δ,2 1-form wavelet ψs . Left: In the Fourier domain the partial integration corresponds to the restriction to the plane τ (a line in 2D) normal to the integration direction Rτ . Middle and 1,δ,2 right: In the spatial domain, the projection of ψs (x) along the Rτ axis (middle) yields its 0,δ,1 restricted counter-part ψs (x1) (right), which is a differential form wavelet in one dimension (cf. Remark 7).

1,δ 1,δ 1 2 = ψs n2 − ψs n1 δ∂Σ(x) dx ∧ dx .

Since, as na¨ıve vectors, (a1, a2) and (a2, −a1) are orthogonal, the above wedge 1,δ product vanishes when ψs is in the normal direction and it is maximized when it is parallel to the boundary (and no metric is required). Hence, as expected, the integral over R2 implements the line integral along ∂Σ. The integral on the right hand side of Eq. 32 is the representation problem

for the cartoon-like function χΣ , which, as discussed above, has been studied extensively in the literature. That a co-exact 1-form wavelet attains the same approximation behavior for the boundary ∂Σ is consistent with the results in [45] where it was shown that polar divergence free wavelets in R2, which correspond to our exact 1-forms, attain the same approximation rates as scalar curvelets.

5.2.2. Fiber Integration of Differential Forms A natural property of differential forms is that the integral of an r-form over a p-dimensional sub-manifold yields an (r − p)-form. In the literature this is sometimes referred to as fiber integration [93]. Our differential form wavelets verify this property for linear sub-manifolds and we will discuss it for the special case p = 1. Thus, we consider the integration along a direction Rτ , R r,ν Rτ r−1,ν ψs −−−−−−→ ψs0 for which the integral of an r-form wavelet is an (r − 1)-form one. Furthermore, r−1,ν n n−1 ψs0 , which can be considered either in R or R , has a closed form expression and s and s0 are closely related, as we will discuss in the following. See Fig. 5 for a depiction for n = 2. ˆr,ν,n In the Fourier domain, the integral along Rτ is equivalent to restricting ψs,a to the plane Pτ with normal τ, analogous to the classical intertwining between restriction and projection in the classical Fourier slice theorem [94]. In the case

36 of volume forms, this was already used in [95] for a local Fourier slice theorem based on polar wavelets. The result carries over to arbitrary polar diﬀerential form wavelets since these are constructed using diﬀerential form basis functions in spherical coordinates. 3 2,δ ˙ 1 2 3 We will demonstrate this in R for ψs ∈ L2(Ωδ, R ). With Def. 3, the 2,δ,3 Fourier representation of ψs , for notational simplicity restricted to the mother wavelet, is i ∂ i 23/2 X ∂ ψˆ2,δ,3(ξ) = − ψˆ (ξ) = − κ y (θ,ˆ φˆ) hˆ(|ξ|) s |ξ| s ∂rˆ |ξ| (2π)3/2 lm lm ∂rˆ lm

Without loss of generality, let τ be along the x3-axis; by the closure of the spherical harmonics bands under rotation, cf. Sec. Appendix A.3, the general case follows by applying a rotation in the spherical harmonics domain using the Wigner-D matrices. The integration along the x3-axis becomes in the Fourier domain the restriction to the ξ1-ξ2 plane and in spherical coordinates this corresponds to θ = π/2. Thus,

23/2 i X ∂ ψˆ2,δ,3(ξ) = − κ y (π/2, φˆ) hˆ(|ξ |) s 3/2 lm lm 1,2 ξ1,2 (2π) |ξ | ∂rˆ1,2 1,2 lm where |ξ1,2| and ∂/∂rˆ1,2 are the restriction of |ξ| and ∂/∂rˆ to the ξ1-ξ2 plane, respectively. Note that these are the natural radial variables in the frequency 2 plane Rb . Expanding the spherical harmonics ylm, cf. Appendix A.3, we obtain 3/2 2 i X ˆ ∂ ψˆ2,δ,3(ξ) = − κ C P m(0) eimφ hˆ(|ξ |) s 3/2 lm lm l 1,2 ξ1,2 (2π) |ξ | ∂rˆ1,2 1,2 lm

3/2 ! 2 i X X ˆ ∂ = − κ C P m(0) eimφ hˆ(|ξ |) 3/2 lm lm l 1,2 (2π) |ξ | ∂rˆ1,2 1,2 m l | {z } ¯ βm

3/2 2 i X ˆ ∂ = − β¯ eimφ hˆ(|ξ |) . (2π)3/2|ξ | m 1,2 ∂rˆ 1,2 m 1,2 ˆ1,δ,2 ˙ 1 1 2 Comparing to the definition of ψs ∈ L2(Ωδ, R ) in Def. 3 and using those ˆ2,δ,3 of the polar wavelets in Eq. 2, we see that the restriction ψs (ξ)|ξ1,2 is the 2 frequency representation of a co-exact 1-form wavelet in the plane R1,2 with ˆ ¯ polar frequency coordinates (φ, |ξ1,2|). The angular localization coefficients βm obtained through the restriction differ from the βm which are commonly used 2 ˆ2,d,3 when one starts in R . From the restriction it is, however, clear that ψs (ξ)|ξ1,2 ˆ will be non-negligible only when the angular window ψs(ξ) is approximately centered on the ξ1-ξ2 plane and that the projected window, characterized by ¯ the βm, will be localized there around its original location, cf. Fig. 4. The localization is hence, in an appropriate sense, preserved under integration. A precise, quantitative analysis of this property is left to future work.

37 5.2.3. Laplace–de Rham operator The Laplace operator acting on differential forms is known as the Laplace–de Rham operator ∆ : Ωr(Rn) 7→ Ωr(Rn). It is defined as [61, Def. 8.5.1] ∆ = dδ + δd = d ? d ? + ? d ? d (33) and, analogous to the usual Laplace operator, it plays a fundamental role in many applications, e.g. [61, Ch. 9]. Since our differential form wavelets are closed under the exterior derivative and the Hodge dual has in the Fourier domain a well defined form, also the Laplace–de Rham operator can be computed in closed form. For example, for a co-exact 2-form wavelet in R3 we have

2,δ 2,δ 2,δ ∆ψs = dδψs + δdψs .

The ﬁrst term vanish by Theorem 3, ii). For the second term we have, using Theorem 3 and Proposition 6, that

2,δ 3,d ? d ? d ψs = ? d ? ψs −1 ˆ0,δ = ? dF |ξ| ψs −1 ˆ1,d = ? F |ξ| ψs −1 2 ˆ2,δ = F |ξ| ψs .

Thus

2,δ −1 2 ˆ2,δ ∆ψs = F |ξ| ψs and the Laplace–de Rham operator is a diﬀerential form of the same type but with a |ξ|2-weight in the radial direction. This is also what one would expect from the scalar Laplace operator. One can check that, up to a sign, this holds r,ν,n for all ψs (x) and we hence have the following the proposition. Proposition 9. The Laplace–de Rahm operator ∆ = dδ + δd of a polar diﬀer- r,ν,n ential form wavelet ψs,a is

r,ν,n −1 |An,r,ν | 2 ˆr,ν,n ∆ψs,a = F (−1) |ξ| ψs,a .

r,ν,n Furthermore, ∆ψs,a is given by the expression for the differential forms wavelets (·+2) in Theorem 2 with the modified radial window, hl (r), obtained through the (spherical) Hankel transform with the additional |ξ|2-weight. The second part of the proposition follows from the fact that the closed form expressions in Theorem 2 are computed using the (inverse) Fourier transform in spherical coordinates, see the proof of the theorem. Remark 10. Proposition 9 shows that the symbol ∆ˆ of the Laplace–de Rahm operator ∆ = dδ +δd is, up to a sign, |ξ|2 and this holds not only for differential form wavelets but for differential forms in general.

38 Dsr

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Figure 6: Decay of the Galerkin projection of the Laplace–de Rahm operator Dqp, as a 2 function of the separation |kq − kp| in R . The decay for isotropic wavelets corresponds to m = 0. For anisotropic wavelets one has a linear combination of the diﬀerent m. Note that the decay does not depend on the wavelet type since our diﬀerential form wavelets use the same frequency window independent of the type of the form.

Important for numerical calculations is the Galerkin projection, r,ν,n r,ν,n Dqp = ∆ψq,a , ψp,a , of the Laplace–de Rahm operator. With the definition of our differential form wavelets in the Fourier domain and Proposition 9 it has a closed form solution. Furthermore, by the compact support of our wavelets in the frequency domain the matrix elements Dqp are nonzero only when min(−1, jp − 1) ≤ jq ≤ jp + 1 and when the orientations align, i.e. tq ≈ tp (assuming compactly supported windows directional windowsγ ˆ(ξ¯); otherwise nonzero has to be replaced by non- negligible). The coefficients Dqp as a function of the separation |kq − kp| are shown in Fig. 6 where it can be seen that these decay fast as |kq − kp| grows. The discrete Laplace–de Rahm operator Dqp is hence well localized in space and frequency by our construction. Example 9. One of the first applications where the importance of differential forms for numerical modeling became clear was electromagnetic theory [3, 4]. The resonant cavity, which we consider in this example, serves thereby as a reference problem [96]. It seeks the electromagnetic field in a simple domain consisting of a perfect conductor containing no enclosed charges. Maxwell’s equations then reduce to an eigenvalue problem for which na¨ıve vector calculus discretizations fail to provide the correct answer [3, 4, 96]. Let B ⊂ R2 be the square domain with side length π. The resonant cavity 1 problem then seeks the electric 1-form field E ∈ Ωδ(B) such that ∆E = δdE = λE in B (34a) i∗E = 0 on ∂B (34b) where the pullback i∗E = 0 describes that the component of E tangential to the boundary vanishes and E is co-exact since the electric field is divergence free in a region free of charges.

39 To be able to apply the polar diﬀerential form wavelets constructed for R2 to the above cavity problem on B, we consider E as a diﬀerential form in R2, analogous to the treatment for Stokes’ theorem in Sec. 5.2.1. This means that the coordinate functions are multiplied by the characteristic function χB and, by abuse of notation, we will not distinguish E and its embedding into R2. Galerkin projection of Eq. 34 using the co-exact 1-form wavelets then yields

DD 1,δ X 1,δEE 1,δ ψp , Eq ∆ ψq = λ ψp ,E q X 1,δ 1,δ Eq ψq , ∆ ψp = λ Ep q X Eq Dqp = λ Ep q where Dqp is the discrete Laplace–de Rahm operator from Remark 10. By linearity of the pullback, the boundary condition becomes in the frame representation

X ∗ 1,δ Eq i ψq = 0 on ∂B. q

1,δ ∗ 1,δ It is hence enforced when the differential form wavelets ψs satisfy i ψs = 0, 1,δ i.e. when the pairing of ψs with the normal vanishes. Fig. 1, right columns in the bottom row, shows that suitably constructed anisotropic polar differential 1,δ 1-form wavelets ψs satisfy this to good approximation. These can hence be used to model the boundary conditions. A numerical investigation of this example is, however, warranted. For example, in practice the boundary conditions do not hold perfectly with the above approach in the finite-dimensional, numerically practical case and there is always leakage of the electric field E, which was embedded in R2, into the exterior domain R2 \B.

5.3. Other Operators on Differential Form Wavelets In the following we will briefly discuss other important operations on differential forms where we do not yet have an elegant realization in Ψec(Rn).

5.3.1. Wedge product r,ν,n As mentioned before, polar differential form wavelets ψs,a are forms in the 0 r,ν,n l,ν ,n sense of the continuous theory. Hence, the wedge product ψs,a ∧ ψs0,a0 is well defined and a (k + l)-form. For numerical calculations one would hope that the nonzero coefficients of the product are sparse, at known locations, and can be computed efficiently. As shown in Proposition 5, the wedge product becomes a convolution in frequency space. Hence, considerable sparsity is lost and precise conclusions about the number and location of nonzero coefficients are not easily established (although the compact support of the wavelets in the frequency domain does provide bounds).

40 In practice we currently use the closed form formulas for the spatial representations of the diﬀerential form wavelets in Theorem 3. With these, the multiplication of the coordinate functions that arises as part of the wedge product can be computed analytically. Although the resulting expressions are rather complicated, they can be combined with numerical quadrature to implement the wedge product. However, we currently have no insight into the sparsity of the coeﬃcients of the product or their decay properties. An alternative to the above approach would be to use the fast transform method [25, 26], i.e. to evaluate the multiplication in the spatial domain. For this, a fast transform is required, which is hence an important direction for future work.

5.3.2. Lie derivative The Lie derivative LX α describes the infinitesimal transport of a differential form α along the vector field X. It hence plays a fundamental role in the description of physical systems using exterior calculus. Using Cartan’s formula, the Lie derivative can be written as

LX α = diX α + iX dα.

While the exterior derivative is well defined for differential form wavelets, the interior product with an arbitrary vector field, which amounts to a multiplication of the coordinate functions, is not. An efficient, general evaluation of the Lie derivative is in Ψec(Rn) hence currently not available. The next example demonstrates that, at least in certain cases, one can resort to the convenient analytic expressions of the differential form wavelets to evaluate the Lie derivative. Example 10. The Euler fluid equation in vorticity form is given by [92, Ch. 1]

∂ζ − L ζ = 0 ∂t ~u

n [ where ~u ∈ Xdiv(R ) is the divergence free velocity vector ﬁeld and ζ = du is 2 2 2 the vorticity, cf. Example 8. In R , vorticity is a volume form, i.e. ζ ∈ Ωd(R ), and using Cartan’s formula we hence have

L~u ζ = d i~u ζ + i~u d ζ = d i~uζ.

In coordinates this equals

∂u ∂ζ ∂u ∂ζ L ζ = − 1 ζ + u + 2 ζ + u dx1 ∧ dx2. ~u ∂x2 1 ∂x2 ∂x1 2 ∂x1

Representing ~u with the divergence free wavelets from [45], which we denote by ~ 2,d ψs, and ζ with the 2-form wavelets ψs we obtain

∂ζq X X D 2,d 2,dE = us ζr L ~ ψr , ψq . ∂t ψs s r

41 The advection or interaction coeﬃcients (cf. [5]) D E Cq = L ψ2,d , ψ2,d sr ψ~s r q can then be computed with a closed form expression, see the supplementary 2,d material, and implementing the reprojection onto ψq using a quadrature rule, cf. [45].

5.3.3. Pullback The pullback η∗α ∈ Ωr(N ) of a differential form α ∈ Ωr(M) by a map η : N → M between two manifolds N , M is a another important operation in the exterior calculus. For linear maps, such as rotation or shear, closed form solutions for the pullback of a differential form wavelet can be derived in the Fourier domain [97], cf. Theorem 3, iv.). For general diffeomorphisms a result by Candèsand Demanet [98, Thm. 5.3] shows that curvelet-like frames are essentially preserved. It would be interesting to extend this result into a numerically practical form. A special case that is of particular relevance are volume preserving diffeomorphisms, which, e.g., describe the time evolution of inviscid fluids. Then the map η can be associated with a unitary operator [99] and Galerkin projection can be used to obtain the representation of η in the wavelet domain.

5.4. Proofs Proof of Theorem 2. We begin with the frame property. The cases of 0- and n- forms are equivalent to the existing results in the literature [32, 19, 33]; for R3 we also provide an alternative, more direct proof in Appendix B. In R2, for 1- form one can use the argument from [45, Proposition 1], i.e. a short calculation reduces the problem to the known scalar case. This argument carries over to 3 2 3 exact forms in R , for which we detail the calculations for L2(Ωd, R ) below. ˙ 1 r n For co-exact forms, which span the homogeneous Sobolev space L2(Ωδ, R ), one can then use the Hodge-Helmholtz decomposition, as we will also explain.

3 2 3 Exact forms in R . Let α ∈ L2(Ωd, R ) be an exact 2-form. By Theorem 1, its Fourier transform is given by ∂ ∂ α = αθ + αφ (35a) b b ∂θˆ b ∂φˆ We want to show that

X DD 2,dEE 2,d X DD 2,dEE 2,d α = α , ψs,1 ψs,1 + α , ψs,2 ψs,2 . (35b) s∈S s∈S Taking the Fourier transform and using Parseval’s theorem for diﬀerential forms we obtain X DD ˆ2,dEE ˆ2,d X DD ˆ2,dEE ˆ2,d αb = αb , ψs,1 ψs,1 + αb , ψs,2 ψs,2 . (35c) s∈S s∈S

42 Considering for the moment only the ﬁrst term and expanding it yields X DD EE X ∂ ∂ ∂ α , ψˆ2,d ψˆ2,d = α + α , ψˆ∗(ξ) ψˆ2,d. b s,1 s,1 bθ ˆ bφ ˆ s ˆ s,1 s∈S s∈S ∂θ ∂φ ∂θ Writing out the inner product with the Hodge dual we obtain Z X DD ˆ2,dEE ˆ2,d X ∂ ∂ ˆ∗ ∂ ∂ ˆ2,d α , ψs,1 ψs,1 = αθ + αφ ∧ ψs (ξ) ∧ ψs,1 . b 3 b ˆ b ˆ ˆ ∂rˆ s∈S s∈S Rb ∂θ ∂φ ∂φ The integral is nonzero only for the ﬁrst term of α, since only then one obtains a volume form. Hence Z X DD ˆ2,dEE ˆ2,d X ˆ∗ ∂ ∂ ∂ ˆ2,d α , ψs,1 ψs,1 = αθ(ξ) ψs (ξ) ∧ ∧ ψs,1 . b 3 b ˆ ˆ ∂rˆ s∈S s∈S Rb ∂θ ∂φ

The integral is now the L2-inner product for scalar functions. Also expanding ˆ2,d ψs,1 on the right hand side of the last equation we can write X DD EE X D E ∂ α , ψˆ2,d ψˆ2,d = α (ξ) , ψˆ (ξ) ψˆ (ξ) . (35d) b s,1 s,1 bθ s s ˆ s∈S s∈S ∂θ | {z } scalar polar wavelet expansion The term with the brace is equivalent to the case of 0-forms, i.e. the known scalar result applies for the representation of the coordinate function and hence the frame property holds for the ﬁrst part ofα ˆ in Eq. 35a. Analogously, one obtains for the second term in Eq. 35c that Z X DD ˆ2,dEE ˆ2,d X ˆ ∂ ∂ ∂ ˆ2,d α , ψs,1 ψs,1 = i αφ(ξ) ψs(ξ) ∧ ∧ ψs,1 b 3 b ˆ ˆ ∂rˆ s∈S s∈S Rb ∂φ ∂θ

X D E ∂ = α (ξ) , ψˆ (ξ) ψˆ (ξ) . bφ s s ˆ s∈S ∂φ | {z } scalar polar wavelet expansion

Co-exact forms in R3. The case of co-exact forms can be handled analogously to the above one by reducing it to a known result for the scalar case. We present here an alternative proof where we exploit the correspondence between co-exact r-forms and exact (r + 1) forms that is established by the Hodge-Helmholtz r,ν,a r,ν decomposition. Let Ψ be the frame operator associated with the ψs,a for ﬁxed ν, r, a. The Gramian is Gr,ν,a = Ψr,ν,a(Ψr,ν,a)∗ and its entries are

r,ν,a r,ν,n r,ν,n Gqp = ψq,a , ψp,a ˙ |ν| . L2 ˙ 1 Using the deﬁnition of the L2 inner product for diﬀerential forms and that r,δ,n r+1,d,n dψq,a = ψq,a we have for co-exact forms that

r,δ,a r,δ,n r,δ,n r+1,d,n r+1,d,n r+1,d,a Gqp = ψq,a , ψp,a ˙ 1 = ψq,a , ψp,a = Gqp , L2 L2

43 i.e. the Gramians for co-exact r-form frame functions and exact (r + 1)-form ones are identical. The lower and upper frame bounds, A and B, respectively, of a frame can be characterized using the Gramian as

A = infha, Gai , a ∈ ran(Ψ), |a| = 1 `2 B = supha, Gai , a ∈ ran(Ψ), |a| = 1 . `2 Since the Gramians are identical and by the Hodge-Helmholtz decomposition d(ran(Ψr,δ)) = ran(Ψr+1,d), the tightness of the frame for exact (r + 1)-forms implies those for co-exact r-forms.

Spatial representation. We will demonstrate the computation of the spatial representation of the diﬀerential form wavelets for one example in R2 and one in R3. An important aspect of the computations is the use of the Jacobi-Anger and Rayleigh formulas that describe the scalar complex exponential in polar and spherical coordinates, respectively, see Appendix A.4. 1,d 2 2 Without loss of generality, we consider the mother wavelet ψ (x) ∈ L2(Ωd, R ) whose deﬁnition in the Fourier domain is given by

X ˆ ∂ ψˆ1,d(ξ) = i β eimθ hˆ(|ξ|) . m ˆ m ∂θ Expanding the form basis functions in polar coordinates into the Cartesian ones we obtain X ˆ ∂ ∂ ψˆ1,d(ξ) = i β eimθ hˆ(|ξ|) sin θˆ − cos θˆ . m ∂ξ1 ∂ξ2 m The trigonometric functions can be written as Fourier series. The Cartesian ˆ1,d ˆ1,d ˆ1,d components ψ1 (ξ) and ψ2 (ξ) of ψ (ξ) then become ! 1 ˆ ˆ X ˆ ∂ ψˆ1,d(ξ) = − e−iθ − eiθ β einθ hˆ(|ξ|) 1 2 n ∂ξ1 n ! 1 X ˆ X ˆ ∂ = − β ei(n−1)θ hˆ(|ξ|) − β ei(n+1)θ hˆ(|ξ|) 2 n n ∂ξ1 n n and ! i ˆ ˆ X ˆ ∂ ψˆ1,d(ξ) = − e−iθ + eiθ β einθ hˆ(|ξ|) 2 2 n ∂ξ2 n ! i X ˆ X ˆ ∂ = − β ei(n−1)θ hˆ(|ξ|) + β ei(n+1)θ hˆ(|ξ|) 2 n n ∂ξ2 n n To obtain the inverse Fourier transform of the 1-forms above we require those of each of the sums in the above formulas. Using the deﬁnition of the inverse

44 Fourier transform for diﬀerential forms in Def. 1 we have, for example, ! X ˆ ∂ F−1 − β ei(n−1)θ hˆ(|ξ|) n ∂ξ1 n Z ! 1 X i(n−1)θˆ ˆ ∂ ihx,ξi ∂ 2 = βn e h(|ξ|) 1 ∧ e 2 dx 2π 2 ∂ξ ∂ξ Rb n ∂ dxq where we immediately only introduced the term of the exponential e ∂ξq that yields a volume form. Changing back to polar coordinates and using the Jacobi- Anger formula to write the scalar complex exponential in polar coordinates we have, ! X ˆ ∂ F−1 − β ei(n−1)θ hˆ(|ξ|) n ∂ξ1 n Z Z 2π 1 X i(n−1)θˆ ˆ X m im(θ−θˆ) ˆ 2 = βn e h(|ξ|) i e Jm(|ξ| |x|) |ξ|d|ξ| dθ dx 2π b+ 0 R n m∈Z Z 2π Z 1 X X m imθ i(n−1)θˆ −imθˆ ˆ ˆ 2 = i βn e e e dθ h(|ξ|) Jm(|ξ| |x|) |ξ|d|ξ| dx 2π 0 b+ n m∈Z R

X n−1 i(n−1)θ 2 = i βn e hm(|x|) dx . n Carrying this out for all terms and using the Fourier transform of the form basis functions in Eq. 15 we obtain 1 X X ψ1,d(x) = σ im+σβ ei(m+σ)θ h (|x|) dx2 1 2 m m+σ σ∈{−1,1} m

i X X ψ1,d(x) = im+σβ ei(m+σ)θ h (|x|) dx1 2 2 m m+σ σ∈{−1,1} m

1,d 1,d 1,d Rewriting this as differential form ψ = ψ2 (x) + ψ1 (x) yields the result in Theorem 3. That the wavelets are real-valued follows from the interaction of σ ∈ {−1, 1} with the factor im+σ that is introduced by the Jacobi-Anger formula. 1,d 1 2 When βm = δm0 one has ψ = h1(|x|)(cos θdx + sin θdx ) = h1(|x|) dr, i.e. a radial differential 1-form field in space. In R3, the calculations proceed analogously to R2 by first representing the spherical form basis functions in Cartesian ones, e.g. ∂/∂φˆ = φˆj∂/∂ξj with φˆj = φˆj(θ,ˆ φˆ), and then using the results of Sec. 4.3. We will present a detailed 1,d,3 calculation for the 1-form basis function ψs,2 . The wavelet is in the frequency domain given by   1 ∂ ∂ ∂ ∂ ˆ1,d,3 ˆ ˆ X ˆj1 j2 ψs,2 = − ψs(ξ) ∧ = iψs(ξ)  φ ∧ rˆ  |ξ| ∂φˆ ∂rˆ ∂ξj1 ∂ξj2 j1

45 By linearity and with the representation of the scalar wavelet from Eq. 4 we obtain   −23j/2 ∂ ∂ ˆ1,d,3 X j,t ˆ −j X ˆj1 j2 ψs,2 = κlm ylm h(2 |ξ|)  φ rˆ ∧  . (2π)3/2|ξ| ∂ξj1 ∂ξj2 l,m j1

3j/2 ˆ1,d,3 2 X X ˆ1,dj1,j2 ˆ −j ∂ ∂ ψs,2 = ψs,2 ylm h(2 |ξ|) ∧ . (2π)3/2|ξ| lm ∂ξj1 ∂ξj2 j1

The spatial representation of the wavelet is given by the inverse Fourier transform in Def. 1. When we also immediately use the Rayleigh formula to expand the scalar complex exponential in spherical coordinates we obtain

3j/2 Z 1,d,3 −2 X X ˆ1,dj1,j2 1 ˆ ˆ ˆ −j ∂ ∂ ψs,2 = 3 ψs,2 lm ylm(θ, φ) h(2 |ξ|) j ∧ j (2π) n |ξ| ∂ξ 1 ∂ξ 2 j1

For each (j1, j2) there is, by the anti-symmetry of the wedge product, only one j3 so that one obtains a volume form, and the necessary reordering of the form basis functions introduces a factor sgn(σj) where j = (j1, j2, j3). In each case we can then change back to spherical coordinates and resolve the integrals over (θ,ˆ φˆ), where the orthonormality of the spherical harmonics applies, and the radial direction |ξ|, where one obtains the modiﬁed spherical Hankel transform (q) hl in Theorem 2. This yields the spatial form for the wavelets listed in the theorem. That the wavelets are real-valued follows from a careful analysis of the spher- ˆ1,ν j1,j2 ˆ2,ν j1 ˆ ical harmonics coeﬃcients ψs,a lm and ψs,a lm. For instance, (φ·rˆ)lm in the above example is nonzero only for even l and the same holds for the κj,t by the l1m1 symmetry of the directional localization windowγ ˆ(θ,ˆ φˆ) that is required for a

46 real-valued wavelet in the scalar case. The product coeﬃcients then preserve the even parity so that the il factor from the Rayleigh formula yields consistently ±1. Proof of Theorem 3. We have: ˆ i) The closure under the exterior derivative holds because d = iiξ only acts on the radial ∂/∂rˆ component. For instance, ˆ ˆ0,δ,3 1 ˆ ∂ ∂ ∂ dψs = iiξ ψs(ξ) ∧ ∧ |ξ| ∂θˆ ∂φˆ ∂rˆ

i ˆ ∂ ∂ ∂ = ψs(ξ) ∧ ∧ |ξ|dˆr |ξ| ∂θˆ ∂φˆ ∂rˆ

ˆ ∂ ∂ = iψs(ξ) ∧ ∂θˆ ∂φˆ

where we used that in spherical coordinate ξ = |ξ|dˆr and that the basis vectors satisfy the duality condition ∂/∂rˆ(dˆr) = 1. The other cases are analogous. That d is a unitary operator on co-exact differential form basis functions follows immediately from the definition of the homogeneous ˙ 1 r n Sobolev space L2(Ωδ, R ). r,δ,n ii) The closure of co-exact differential form wavelets ψs,a under the co-differential holds since these are co-exact differential forms in the sense of the continuous theory. iii) This can be checked by direct calculations. iv) This follows from the closure of scalar polar wavelets under rigid body transformations [97] and the fact that we use frame functions in spherical coordinates that are compatible with the scalar window.

6. Conclusion

We introduced Ψec(Rn), a wavelet-based discretization of exterior calculus. r,ν,n Its central objects are polar differential r-form wavelets ψs,a that provide tight ˙ |ν| r n frames for the spaces L2 (Ων , R ) and that satisfy important properties of Car- tan’s exterior calculus, such as closure under the exterior derivative. In contrast to existing discretizations, which typically construct a discrete analog of the k,ν,n de Rahm complex, the ψs,a are bona fide differential forms in the sense of the continuous theory. All operations on differential forms are hence naturally defined. Finite closure and the computational efficiency of the discretization hence become the questions of interest. The present work provides, in our opinion, a first step towards a more com- plete theory. For example, except when n = 0 or r = n and the existing results for cartoon-like functions apply, we did not establish approximation rates for

47 polar differential form wavelets. Similarly, for the Laplace–de Rahm operator we are missing decay estimates for its Galerkin projection Dqp, which are, for example, necessary for error bounds for Example 9. We also did not yet present numerical experiments. An implementation of the differential form wavelets is available, cf. Fig. 1 and Fig. 4, and numerical results for example applications will be presented in a forthcoming publication. Since our wavelets have non-compact support in space, the efficiency of numerical calculations depends critically on the decay of the radial window. It should hence be optimized, cf. [100], or an ansatz should be developed that yields compactly supported wavelets. Then also a fast transform would be available. Our work can be extended in various directions. We defined our differential form wavelets only for n = 2, 3 but the construction carries over to n > 3. One then would require more generic proofs then the ones currently used by us that typically proceed explicitly with the individual cases. However, with the tools introduced in Sec. 4 this should be possible. It would also be interesting to extend the functional analytic setting of our work. Troyanov [74] considers, r n for example, Lp(Ω , R ) and spaces with variable regularity, for example in the sense of Hölder,which would be relevant for applications. Various extensions of our treatment of Stokes’ theorem are also possible. We only considered smooth manifolds but using the tools of geometric measure theory [101] it should be possible to establish a similar result for the non-smooth case. Given that we are limited to an extrinsic treatment of manifolds, it would also be interesting to explore if our construction can be combined with those by Berry and Giannakis [56] who use the eigenfunctions of the Laplace–Beltrami operator to define a spectral exterior calculus on arbitrary manifolds. A special case of particular relevance is the sphere S2 where one still has an extensive analytic theory available and where geophysical fluid dynamics provides an important application. The construction of Ψec on S2 and numerical results on the simulation of the shallow water equation are presented in [46].

Acknowledgements

I would like to thank Mathieu Desbrun and Tudor Ratiu for helpful discussions. The work also beneﬁted from discussions at the Workshop on Mathe- matics of Planet Earth organized by Darryl Holm in June 2019. The insightful suggestions by the anonymous reviewer are also gratefully acknowledged. Con- tinuing support by Eugene Fiume over many years also helped to make this work possible.

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Appendix A. Conventions

Appendix A.1. Notation We denote the norm in Rn by | · | and those in a function space by k · k. Unit vectors are written asx ¯, i.e. for x ∈ Rn we havex ¯ = x/|x|. We will not always distinguish between a unit vector and its corresponding coordinates in polar or spherical coordinates and depending on the contextx ¯ might thus be a geometric vector or its spherical coordinates (θ, φ). We use standard spherical coordinates with latitude θ ∈ [0, π] and longitude φ ∈ 2π. The standard area form in polar coordinates is r dθ dr and in spherical coordinates the volume form is r2 sin θ dθ dφ dr. For example, the integral of f : R3 → R takes in spherical coordinates thus the form Z Z Z f(x) dx = f(r, θ, φ) r2 sin θ dθ dφ dr 3 + 2 R R S Z Z π Z 2π = f(r, θ, φ) r2 sin θ dθ dφ dr + R θ=0 φ=0 where R+ means that the radial variable |x| is integrated over the positive real line. We will use similar notation whenever confusion might arise otherwise.

Appendix A.2. The Fourier transform and Homogeneous Sobolev Spaces n n The scalar unitary Fourier transform of a function f ∈ L2(R ) ∩ L1(R ) is 1 Z F(f)(ξ) = fˆ(ξ) = f(x) e−ihx,ξi dx n/2 (2π) n R with inverse transform 1 Z F−1(f)(ξ) = f(x) = fˆ(ξ) eihξ,xi dξ. n/2 (2π) n Rb ˙ 1 n The homogeneous Sobolev space L2(R ) is deﬁned as n o ˙ 1 n n L2(R ) = f ∈ L1(R ) [f]0 ˙ 1 < ∞ L2 with the norm being those induced by the inner product [102, Ch. 1.2.1] Z ˆ 2 hf, giL˙ 1 = f(ξ)g ˆ(ξ) |ξ| dξ 2 n Rb

57 and [f]0 denoting the co-sets of functions modulo constant polynomials [75, Ch. II.6]. As is customary, the co-sets are usually omitted in the notation. The so ˙ 1 n deﬁned space L2(R ) is a Hilbert space [75, Ch. II.6].

Appendix A.3. Spherical harmonics The analogue of the Fourier transform on the sphere is the spherical har- 2 monics expansion. For any f ∈ L2(S ) it is given by

∞ l ∞ l X X X X f(ω) = hf(η), ylm(η)i ylm(ω) = flm ylm(ω) l=0 m=−l l=0 m=−l

2 where h·, ·i denotes the standard (complex) L2 inner product on S given by

Z π Z 2π hf(ω), g(ω)i = f(θ, φ) g∗(θ, φ) sin θ dθdφ. θ=0 φ=0 The spherical harmonics basis functions are given by

m imφ ylm(ω) = ylm(θ, φ) = Clm Pl (cos θ) e

m where the Pl (·) are associated Legendre polynomials and Clm is a normalization constant so that the ylm(ω) are orthonormal over the sphere. The spherical harmonics addition theorem is

l 4π X P (¯x · x¯ ) = y (ω ) y∗ (ω ) (A.1) l 1 1 2l + 1 lm 1 lm 2 m=−l

3 where ω1 are the spherical coordinates for the unit vectorx ¯1 ∈ R and ω2 are 3 those forx ¯2 ∈ R . It follows immediately from the above formula that Pl(¯x1 ·x¯1) is the reproducing kernel for the space Hl spanned by all spherical harmonics of ﬁxed l. The Hl are closed under rotation, i.e. f ∈ Hl ⇒ Rf ∈ Hl for R ∈ SO(3) and the action is given by (Rf)(ω) = f(RT ω). In the spherical m0 harmonics domain the rotation is represented by Wigner-D matrices Wlm (R) that act as

m0 flm(R) = Dlm flm0 where the flm(R) are the spherical harmonics coeﬃcients of the rotated signal. 2 0 When R describes a rotation of the North pole to λ ∈ S , then Dlm takes a simple form,

r 4π D0 (λ) = y∗ (λ). lm 2l + 1 lm In contrast to the Fourier series, where the product of two basis functions eim1θ and eim2θ is immediately given by another Fourier series function,

58 ei(m1+m2)θ, for spherical harmonics the product is not diagonal and instead characterized by Clebsch-Gordon coeﬃcients Cl,m . In particular, the product l1,m1;l2,m2 of two functions on the sphere is in the spherical harmonics domain given by ! X X X f · g(ω) = f g Gl,m y (ω) l1m1 l2m2 l1,m1;l2,m2 lm lm l1m1 l2m2 | {z } (f · g)lm where the spherical harmonics product coeﬃcients Gl,m are l1,m1;l2,m2 Z Gl,m ≡ y (ω) y (ω) y∗ (ω) dω l1,m1;l2,m2 l1m1 l2m2 lm S2 s (2l + 1)(2l + 1) = 1 2 Cl,0 Cl,m . 4π(2l + 1) l1,0;l2,0 l1,m1;l2,m2

The Clebsch-Gordon coeﬃcients Cl,m are sparse and nonzero only when l1,m1;l2,m2

m = m1 + m2, that is the m parameter is superﬂuous but conventionally used, and

l1 + l2 − l ≥ 0

l1 − l2 + l ≥ 0

−l1 + l2 + l ≥ 0.

Appendix A.4. Fourier Transform in Polar and Spherical Coordinates In the plane, the Fourier transform can also be written in polar coordinates by using the Jacobi-Anger formula [103],

ihξ,xi X m im(θ−θˆ) e = i e Jm(|ξ| |x|), m∈Z that relates the complex exponential in Euclidean and polar coordinates. In the above equation, Jm(z) is the Bessel function of the ﬁrst kind and (θ, |x|) and (θ,ˆ |ξ|) are polar coordinates for the spatial and frequency domains, respectively. The analogue of the Jacobi-Anger formula in three dimensions is the Rayleigh formula, ∞ ihξ,xi X l ¯ e = i (2l + 1) Pl ξ · x¯ jl(|ξ| |x|) l=0

∞ l X X l ¯ = 4π i ylm ξ ylm x¯ jl(|ξ| |x|) l=0 m=−l where jl(·) is the spherical Bessel function and the second line follows by the spherical harmonics addition theorem in Eq. A.1.

59 3 Appendix B. Admissibility of Polar Wavelets for L2(R ) Proof of Proposition 2. Our proof will be for a scale and directionally discrete frame that is continuous in the spatial domain, i.e. where all translations in R3 are considered. Since our window functions are bandlimited, the result carries over to the discrete case using standard arguments based on the Shannon sampling theorem, see for example [22, Sec. 4] or [32, Sec. IV]. We want to show

∞ Tj X X u = u ? ψj ? ψj j=−1 t=1

3 for u ∈ L2(R ). Taking the Fourier transform of both sides we have

∞ Tj ∞ Tj X X ˆ ˆ X X ˆ∗ ˆ uˆ = (û ψj) ψj =u ˆ ψj ψj j=−1 t=1 j=−1 t=1 and it thus suffices to show that the scalar window functions satisfy the Calderón admissibility condition

∞ Tj X X ˆ 2 3 |ψj| = 1 , ∀ξ ∈ R . j=−1 t=1 With the deﬁnition of the window functions in Eq. 4 one obtains

∞ Tj Tj X X X X X X 2 |ψˆ |2 = κj,t y ξ¯ κj,t y∗ ξ¯ hˆ(2−j|ξ|) . j l1m1 l1m1 l2m2 l2m2 j=−1 t=1 j t=0 l1,m1 l2m2 Assuming the radial windows satisfy the Calder´oncondition, i.e.

∞ X −j 2 hˆ(2 |ξ|) = 1 j=−1 the proposition holds when the product of the angular part evaluates to the ¯ identity for every band j and every direction ξ. Since y00 is the constant function, this means that for every j the projection of the angular part in the above equation onto spherical harmonics has to satisfy

Tj * + X X X δ δ = κj,t y ξ¯ , κj,t y∗ ξ¯ , y (ξ¯) . l,0 m,0 l1m1 l1m1 l2m2 l2m2 lm t=0 l1,m1 l2,m2 Rearranging terms we obtain

Tj X X X δ δ = κj,t κj,t∗ y ξ¯y∗ ξ¯ , y∗ (ξ¯) . l,0 m,0 l1m1 l2m2 l1m1 l2m2 lm t=0 l1,m1 l2,m2

60 The product of two spherical harmonics projected into another spherical harmonic is given by the product coeﬃcients Gl,m introduced in Appendix l1,m1;l2,m2 A.3. We can hence write

Tj X X X δ δ = κj,t κj,t Gl,m1−m2 . l,0 m,0 l1m1 l2m2 l1,m1;l2,−m2 t=0 l1,m1 l2,m2

Collecting the κj,t into vectors we obtain the condition in the proposition. limi