Split-Quaternion Harmonic Analysis

Introduction

This article treats the harmonic analysis of the split-quaternion algebra. It identifies the additive group and its dual, develops the Fourier transform of the group, proves the convolution theorem, introduces the algebra-valued transform and the vanishing-determinant issue on the null cone, treats distributions and the transform of the fundamental solution, and compares the situation with the quaternion and split-complex transforms.

The split-quaternion algebra, its split-quaternion norm and its units are assumed from Split-Quaternion Algebra and Split-Quaternion Norm and Invertibility; the exponential and its invertibility from Split-Quaternion Elementary Functions; the operators, their characteristic variety and the failure of elliptic regularity from Split-Quaternion Analysis; the fundamental solution and its support on the cone from Split-Quaternion Integration; and the metric and the forms from Split-Quaternion Geometry. The harmonic analysis of Euclidean space is that of Fourier Analysis on Euclidean Spaces, the analysis of the abelian group and its dual that of Harmonic Analysis on Groups, the distributions that of Distributions and Fundamental Solutions, the division-algebra transform that of Quaternion Harmonic Analysis, and the two-dimensional hyperbolic transform that of Split-Complex Harmonic Analysis. Nothing physical is invoked.

The Additive Group and Its Dual

Theorem (The Group and Its Dual). The additive group of the algebra, identified with $\mathbb{R}^4$ through the coordinates $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$, is a locally compact abelian group, with the Lebesgue measure as its Haar measure. Its dual is again $\mathbb{R}^4$, identified with the frequency vectors $\xi = \alpha + \beta e_1 + \gamma e_2 + \delta e_3$, the pairing being the Euclidean scalar product of the coordinate vectors and the characters being the complex-valued functions

$$ \chi_\xi(\tilde q) = e^{2\pi \mathrm{i}\langle \tilde q,\xi\rangle}, \qquad \langle \tilde q,\xi\rangle = q_0\alpha + q_1\beta + q_2\gamma + q_3\delta , $$

Pontryagin duality holds, and the dual group is equipped with the dual Haar measure.

Proof. The additive group of a finite-dimensional real vector space with its Euclidean topology is locally compact and abelian, and the character group of $\mathbb{R}^n$ is $\mathbb{R}^n$ with the displayed characters; this is the standard duality of Harmonic Analysis on Groups, §Characters and the Dual Group.

Remark (The Complex Unit Is Scalar). The symbol $\mathrm{i}$ in the characters is the usual complex unit, a scalar; it is not an element of the algebra, and no element of the algebra is ever used as the imaginary unit of the transform. The algebra enters only through the values of the functions transformed and through the products of those values.

Remark (The Corpus Convention). The characters and the transform are taken with the sign convention of Quaternion Harmonic Analysis, §The Quaternion Characters, so that the pairing is the Euclidean one on $\mathbb{R}^4$ and the transform of a derivative carries the factor $-2\pi\mathrm{i}$. In the quaternion case the algebra-valued characters $e^{2\pi\omega\langle \tilde q,\xi\rangle}$ with $\omega^2 = -1$ lie on the unit sphere and are bounded; here the corresponding kernel $e^{-\tilde q\xi}$ is bounded only when $\operatorname{Sc}(\tilde q\xi) = 0$, and its split-quaternion norm is $e^{-2\operatorname{Sc}(\tilde q\xi)}$ in general, because the form is indefinite. The boundedness of the characters in the definite case and its failure in the split case is the same dichotomy as the definiteness of the form.

Corollary (The Space of Functions). The algebra-valued functions with square-integrable components form the Hilbert space

$$ L^2(\mathbb{H}_{\mathrm{s}}, \mathbb{H}_{\mathrm{s}}) \cong \big(L^2(\mathbb{R}^4)\big)^4 , $$

with the inner product $\langle f,g\rangle = \int \operatorname{Sc}(f g^{\natural})$, and the algebra acts pointwise on it by left and right multiplication.

Proof. The identification is componentwise, and the scalar product is that of four real functions expressed in the basis.

The Fourier Transform of the Group

Definition. For $f \in L^1(\mathbb{H}_{\mathrm{s}},\mathbb{H}_{\mathrm{s}})$ the Fourier transform is

$$ \hat f(\xi) = \int_{\mathbb{H}_{\mathrm{s}}} f(\tilde q)\, e^{2\pi \mathrm{i}\langle \tilde q,\xi\rangle}\,\mathrm{d}\tilde q \ \in \mathbb{H}_{\mathrm{s}} , $$

the transform of the four scalar components of $f$.

Theorem (Inversion and Plancherel). For $f$ in the Schwartz class of algebra-valued functions,

$$ f(\tilde q) = \int_{\mathbb{H}_{\mathrm{s}}} \hat f(\xi)\, e^{-2\pi \mathrm{i}\langle \tilde q,\xi\rangle}\,\mathrm{d}\xi , $$

the transform extends to a unitary operator on $L^2(\mathbb{H}_{\mathrm{s}},\mathbb{H}_{\mathrm{s}})$ up to the usual normalisation, and the Riemann–Lebesgue lemma holds. All of this is the classical Fourier analysis of $\mathbb{R}^4$ applied to each of the four components.

Proof. The scalar theory of Fourier Analysis on Euclidean Spaces applies componentwise; the unitarity is the Plancherel theorem in each component.

Corollary (The Non-Commutativity Plays No Role Here). The group transform sees only the abelian additive structure of the algebra: the non-commutativity of the product and the zero divisors do not enter the statements of inversion and Plancherel, because the characters are scalar and commute with everything. They enter as soon as a product of algebra-valued functions is taken, that is in the next two sections.

Proof. The characters are central; the transform is componentwise.

Convolution and the Convolution Theorem

Definition. The convolution of two algebra-valued functions is

$$ (f * g)(\tilde q) = \int_{\mathbb{H}_{\mathrm{s}}} f(\tilde p)\,g(\tilde q-\tilde p)\,\mathrm{d}\tilde p , $$

with the product of the algebra inside the integral.

Theorem (The Convolution Theorem). For $f, g \in L^1(\mathbb{H}_{\mathrm{s}},\mathbb{H}_{\mathrm{s}})$ the convolution is well defined almost everywhere, and

$$ \widehat{f * g}(\xi) = \hat f(\xi)\,\hat g(\xi), $$

the product being the product of the algebra.

Proof. Substituting $\tilde r = \tilde q-\tilde p$ in the transform of $f*g$ and using the multiplicativity of the characters, which are scalar, separates the double integral into the product of the two transforms; the algebra product factors out because the character is central.

Corollary (The Transform Is an Algebra Homomorphism). The Fourier transform carries convolution to the pointwise product of the algebra and the direct sum to the sum, both products taken in the same order; it is therefore an isomorphism of the convolution algebra onto the algebra of algebra-valued functions with the pointwise product. The convolution is associative, because the algebra product is, but it is not commutative: $\widehat{f*g} = \hat f\hat g$ while $\widehat{g*f} = \hat g\hat f$, and the two differ as soon as the values of the transforms fail to commute. The transform is an isomorphism onto the pointwise-product algebra, not onto a commutative algebra, and the non-commutativity of the algebra is exactly what is transported.

Proof. The theorem and the inversion formula give the isomorphism; for the failure of commutativity take $f = e_1k$ and $g = e_2k$ with a fixed integrable $k$ of positive square integral: then $(f*g)(0) = e_3\int k(\tilde p)k(-\tilde p)\,\mathrm{d}\tilde p$ and $(g*f)(0) = -e_3\int k(\tilde p)k(-\tilde p)\,\mathrm{d}\tilde p$, which differ. Associativity is $\widehat{(f*g)*h} = (\hat f\hat g)\hat h = \hat f(\hat g\hat h)$.

Approximate Identities and Young's Inequality

Definition. An approximate identity is a family $(k_\varepsilon)_{\varepsilon>0}$ of integrable algebra-valued functions with $\int k_\varepsilon = 1$, with $\sup_\varepsilon \|k_\varepsilon\|_1$ finite and with the mass concentrating at the origin as $\varepsilon \to 0$; the standard choice is $k_\varepsilon(\tilde q) = \varepsilon^{-4}k(\tilde q/\varepsilon)$ for a fixed integrable $k$ with $\int k = 1$.

Theorem (Young's Inequality and Convergence). For $f \in L^p$ and $k \in L^1$,

$$ \|f * k\|_p \leq C\,\|f\|_p\,\|k\|_1 , $$

with the constant $C = \sqrt2$ supplied by the submultiplicative law of the Euclidean norm as in Split-Quaternion Analysis, §The Metric Structure; and $f * k_\varepsilon \to f$ in $L^p$ as $\varepsilon \to 0$ for $1 \leq p < \infty$, so the $L^1$-convolution has bounded approximate identities.

Proof. Young's inequality is the classical one, the algebra-valued integrand being estimated by the submultiplicative norm; the convergence of the approximate identity uses the translation invariance of the Lebesgue measure and the continuity of translation in $L^p$, both componentwise.

The Algebra-Valued Transform and the Vanishing Determinant

The transform that uses the algebra itself as the kernel is the analogue of the quaternionic and Clifford transforms; it is not the group transform, and its inversion is where the zero divisors appear.

Definition. For a suitable function $f$ the algebra-kernel transform is

$$ \mathcal{F}f(\xi) = \int_{\mathbb{H}_{\mathrm{s}}} e^{-\tilde q\xi}\, f(\tilde q)\,\mathrm{d}\tilde q , \qquad \tilde q\xi \text{ the algebra product}, $$

the exponential being that of Split-Quaternion Elementary Functions.

Theorem (The Kernel Is Never Singular). The kernel $e^{-\tilde q\xi}$ is invertible for all $\tilde q$ and all $\xi$, with

$$ N\big(e^{-\tilde q\xi}\big) = e^{-2\operatorname{Sc}(\tilde q\xi)} > 0 , $$

so the exponential kernel itself is never a zero divisor, and the transform is well defined and bounded on $L^1$ for every frequency.

Proof. Split-Quaternion Elementary Functions, §The Exponential, applied to the element $-\tilde q\xi$.

Theorem (The Inversion Degenerates on the Null Cone). The kernel $e^{-\tilde q\xi}$ is not a character of the group: it is algebra-valued and not multiplicative in $\tilde q$. The inversion of $\mathcal{F}$ requires, in its natural derivation by integration by parts in the frequency variable, the inverse of the difference variable $\tilde q-\tilde p$, and

$$ (\tilde q-\tilde p)^{-1} = \frac{\overline{\tilde q-\tilde p}}{N(\tilde q-\tilde p)} $$

exists exactly when $N(\tilde q-\tilde p) \neq 0$, that is off the null cone. On the null cone the reciprocal does not exist, the derivation of the inversion formula fails, and the transform is not invertible by a kernel of the same shape.

Proof. The inverse of an element is $u^{\natural}/N(u)$ by Split-Quaternion Norm and Invertibility, §The Invertibility Criterion, which likewise gives that $N(u) = 0$ exactly on the zero divisors. The non-multiplicativity of the kernel is immediate from the non-commutativity: $e^{-(\tilde q+\tilde p)\xi} \neq e^{-\tilde q\xi}e^{-\tilde p\xi}$ unless $\tilde q$ and $\xi$ commute, by Split-Quaternion Elementary Functions, §Non-Commutativity and the One-Variable Case.

Theorem (The Vanishing Determinant, Stated Precisely). Let $D = e_1\partial_{q_1} + e_2\partial_{q_2} + e_3\partial_{q_3}$ be the vector operator and let the transform be the group transform of the preceding section, whose linearity allows differentiating under the integral. Then

$$ \widehat{Df}(\xi) = -2\pi \mathrm{i}\;\xi\, \hat f(\xi), $$

so that the symbol of the operator is the frequency vector $\xi \in V$; the symbol is invertible exactly when $\xi$ is a unit, that is exactly when $N(\xi) \neq 0$, and it degenerates on the null cone, the zero divisor set. The singular set of the symbol is the characteristic variety of the operator, in agreement with Split-Quaternion Analysis, §The Differential Operators.

Proof. Integration by parts in each coordinate gives $\widehat{\partial_i f} = -2\pi\mathrm{i}\xi_i\hat f$, and the generators multiply the scalar-valued transform centrally; the identification of the invertible frequencies with $N \neq 0$ is the invertibility criterion of Split-Quaternion Norm and Invertibility.

Corollary (The Transform of the Fundamental Solution Is Singular on the Cone). Let $E_D$ be the fundamental solution of $D$ of Split-Quaternion Integration, §The Fundamental Solution. Then, with the principal value of the reciprocal,

$$ \hat E_D(\xi) = -\frac{1}{2\pi\mathrm{i}}\,\xi^{-1} = \frac{1}{2\pi\mathrm{i}}\,\frac{\xi}{N(\xi)} , $$

a distribution whose singular support is exactly the null cone. The vanishing determinant $N(\xi) = 0$ is therefore not a technical nuisance: it is the support of the transformed fundamental solution, and it is the reason the fundamental solution is singular on the cone rather than at a point.

Proof. From $\widehat{DE_D} = -2\pi\mathrm{i}\xi\hat E_D = \hat\delta = 1$ and $\xi^{-1} = -\xi/N(\xi)$ for $\xi \in V$; the support statement is then the standard computation of the principal value of $1/N$ on an indefinite quadratic form, as in Distributions and Fundamental Solutions.

Distributions

Theorem (Tempered Distributions). The Fourier transform extends to an isomorphism of the space of tempered algebra-valued distributions onto itself, and it exchanges multiplication by the algebra-valued functions with convolution. The fundamental solution $E_D$ is tempered, being supported in a closed cone and of polynomial growth, so its transform exists and is the distribution displayed above.

Proof. The scalar theory of Distributions and Fundamental Solutions applies componentwise; the temperedness of the fundamental solution follows from its homogeneity of degree $-2$ and its support on the cone.

Corollary (The Failure of Elliptic Inversion and Its Frequency Location). The inversion of the transform of a solution of $Df = g$ is obstructed exactly on the null cone: the frequency equation $\hat f = -(2\pi\mathrm{i}\xi)^{-1}\hat g$ cannot be solved by a smooth symbol on the whole frequency space, and any solution must have its singularities concentrated on the characteristic variety. This is the frequency-space form of the failure of the elliptic theory and of the cone support of the fundamental solution.

Proof. The symbol is non-invertible on the null cone by the vanishing-determinant theorem, so the inverse symbol is not smooth there and the multiplication by it is not a pseudodifferential operation of the standard elliptic type.

The Symbol and Its Degeneracy

Theorem (The Symbol and the Singular Frequencies). The symbol of the vector operator at frequency $\xi$ is the frequency $\xi$ itself, and the symbol is invertible exactly when $N(\xi) \neq 0$; the singular frequencies are the null cone, which is the zero divisor set of the algebra.

Proof. The operator has constant coefficients, so its symbol is the frequency vector, and the symbol is invertible exactly when the frequency is a unit, that is when $N(\xi) \neq 0$ by the invertibility criterion; the set $\{N = 0\}$ is the null cone, the zero divisor set.

The degeneracy of the harmonic analysis on the null cone is therefore read in the algebra alone, as the non-invertibility of the frequency: the vanishing of the determinant of the symbol is the vanishing of the split-quaternion norm, and no further object is needed to locate it.

Comparison with the Quaternion and Split-Complex Transforms

	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{D}$
additive group	locally compact abelian	locally compact abelian	locally compact abelian
group transform	four-dimensional Fourier transform	four-dimensional Fourier transform	two-dimensional Fourier transform
convolution theorem	holds	holds	holds
algebra-kernel transform	quaternionic Fourier transform, invertible symbol	algebra-kernel transform, symbol degenerate on the null cone	hyperbolic transform, degenerate on the isotropic lines
symbol $\xi$	invertible for every $\xi \neq 0$	invertible iff $N(\xi) \neq 0$	invertible iff $N(\xi) \neq 0$
transformed fundamental solution	singular at the origin	singular on the light cone	singular on the two isotropic lines

The quaternion column is the content of Quaternion Harmonic Analysis and the split-complex column that of Split-Complex Harmonic Analysis. The decisive difference is that the quaternion norm vanishes only at the origin, so its symbol is invertible at every nonzero frequency and the quaternionic transform has no vanishing-determinant problem; in the split case the split-quaternion norm vanishes on a cone, of real dimension two in $V$, and the transformed fundamental solution is supported there. The eight-dimensional relative $\mathbb{H}_{\mathbb{D}}$ is a later system of Part V, treated under Split-Biquaternions, and nothing of it is used here.

Summary

The additive group of the split-quaternion algebra is a locally compact abelian group, identified with $\mathbb{R}^4$, with dual $\mathbb{R}^4$ and scalar characters $\chi_\xi(\tilde q) = e^{2\pi\mathrm{i}\langle \tilde q,\xi\rangle}$. The Fourier transform of the group is the fourfold scalar transform applied to the components, with inversion and Plancherel; the convolution theorem holds with the algebra product, because the characters are scalar, so the transform is an isomorphism of the convolution algebra onto the pointwise product algebra; the convolution is associative because the algebra product is, and it is not commutative, because the algebra product is not.

The algebra-kernel transform uses the exponential of the algebra as kernel. The kernel is never a zero divisor, since $N(e^u) = e^{2\operatorname{Sc}u} > 0$, but it is not a character, and the inversion formula requires the reciprocal of the difference variable, which exists exactly off the null cone; the transform degenerates there. For the vector operator the symbol is the frequency vector $\xi$, invertible exactly when $N(\xi)\neq0$; the singular frequencies form the null cone, which is the characteristic variety of the operator and the zero divisor set. The transformed fundamental solution is $\hat E_D = -(2\pi\mathrm{i}\xi)^{-1} = \xi/(2\pi\mathrm{i}N(\xi))$, a distribution whose singular support is exactly the null cone, and the vanishing determinant is thus the frequency-space explanation of the cone-supported fundamental solution and of the failure of the elliptic theory. The symbol is the frequency $\xi$ itself, and it is singular exactly on the null cone.

The comparison with the quaternion case is sharp: there the split-quaternion norm vanishes only at the origin, the symbol is invertible at every nonzero frequency, and the transformed fundamental solution is singular at a point; in the split case the split-quaternion norm vanishes on a cone and every degeneracy is located there. The split-complex case is the two-dimensional model of the same phenomenon, with the two isotropic lines in place of the cone. The eight-dimensional relative is a later system of Part V, named only.

The two transforms of the article are therefore to be kept apart: the group transform is the harmonic analysis of the additive group, complete with inversion, Plancherel and the convolution theorem, and it is blind to the zero divisors; the algebra-kernel transform is the non-commutative transform of the system, and every one of its difficulties is located on the null cone, where the symbol ceases to be invertible. The same cone is the zero divisor set of the algebra, the characteristic variety of the vector operator and the support of the transformed fundamental solution, and it is the single object around which the analysis of this system is organised.

Summary of Notation

Symbol	Meaning	Article
$\chi_\xi(\tilde q) = e^{2\pi\mathrm{i}\langle \tilde q,\xi\rangle}$	the scalar characters of the additive group	this article
$\hat f(\xi)$	the group Fourier transform	this article
$f * g$	the convolution with the algebra product	this article
$\widehat{f*g} = \hat f\hat g$	the convolution theorem	this article
$\mathcal{F}f(\xi) = \int e^{-\tilde q\xi}f$	the algebra-kernel transform	this article
$N(e^{-\tilde q\xi}) = e^{-2\operatorname{Sc}(\tilde q\xi)} > 0$	the kernel is never a zero divisor	Split-Quaternion Elementary Functions
$\xi^{-1}$, $N(\xi)$	the symbol and its determinant	Split-Quaternion Norm and Invertibility
$\hat E_D = -(2\pi\mathrm{i}\xi)^{-1} = \xi/(2\pi\mathrm{i}N(\xi))$	the transformed fundamental solution, singular on the null cone	Split-Quaternion Integration
null cone	the singular frequency set and the zero divisor set	Split-Quaternion Zero Divisors