Mathematical foundations of idempotent-generated multidimensional analytic signals: existence of extended Hilbert operators, necessity of multiplier idempotency conditions, and boundary-value structure

Sangston (2024) proposed a unifying framework in which a multidimensional analytic signal is generated by an idempotent-valued Fourier multiplier in a geometric algebra, recovering Hahn's single-orthant signal, Bülow and Sommer's hypercomplex signal, and Felsberg and Sommer's monogenic signal as special cases. The construction is deliberately formal and example-driven: the paper gives only sufficient algebraic conditions for a multivector Fourier multiplier to generate an idempotent, introduces the extended Hilbert operator formally as the inverse Fourier transform of the multiplier, and notes that an appropriate convolutional operator need not exist for an arbitrary idempotent. This problem asks for the rigorous mathematical characterization of that framework: necessary and sufficient conditions on multivector Fourier multipliers for idempotency; algebraic and analytic conditions under which the extended Hilbert operator exists on a stated function space and admits a spatial convolution (singular-integral) kernel; the identification of the multidimensional singular-integral operators arising in the framework; and the boundary-value (monogenic, Riemann-Hilbert-type) problems that generate the generalized analytic signals. No resolution of any of these questions is known in the literature to date.

Background

For a real one-dimensional signal $f(x)$, the Hilbert transform $H[f](x)=\frac{1}{\pi}\mathrm{PV}\int\frac{f(t)}{t-x}\,dt$ (a principal-value singular integral) has Fourier multiplier $-i\,\mathrm{sgn}(\omega)$, and the analytic signal $f_A=f+iHf$ has a one-sided spectrum, from which instantaneous amplitude, phase, and frequency are read off. Equivalently, $\mathcal{F}[f_A](\omega)=2\psi_{\mathcal{H}_1}(\omega)\mathcal{F}[f](\omega)$ where $\psi_{\mathcal{H}_1}(\omega)$ is the projection onto nonnegative frequencies; crucially $\psi_{\mathcal{H}_1}$ is an idempotent, i.e. a quantity that squares to itself.

Several inequivalent extensions of this construction to signals $f(\boldsymbol{x})$ on $\mathbb{R}^N$ exist. Hahn's single-orthant signal multiplies the spectrum by a product of one-dimensional sign functions. Bülow and Sommer's hypercomplex signal uses a quaternionic (Clifford) Fourier transform so that the components of the transformed signal are partial Hilbert transforms. Felsberg and Sommer's monogenic signal is built from the Riesz transforms $R_1,\dots,R_N$ — the vector-valued singular integrals with Fourier multipliers $-i\omega_j/|\boldsymbol{\omega}|$, the standard higher-dimensional analogues of the Hilbert transform — and is tied to monogenic functions, i.e. multivector-valued null-solutions of the Dirac operator $D=\sum_k e_k\partial_{x_k}$ (in the monogenic-signal construction, on $\mathbb{R}^{N+1}$ with coordinates $(x_0,\boldsymbol{x})$, so that $D=\partial_{x_0}+\sum_{k=1}^{N}e_k\partial_{x_k}$ and the boundary of the upper half-space $\mathbb{R}^{N+1}_{+}$ is the signal domain $\mathbb{R}^N$), which play the role in Clifford analysis that holomorphic functions play in one complex variable. In one dimension the Hilbert transform is essentially the only singular integral of the Calderon-Zygmund type (a principal-value convolution operator whose kernel is homogeneous of degree $-N$ and satisfies the standard size and smoothness conditions); in higher dimensions the Riesz transforms and their Clifford generalizations are the corresponding examples, and Clifford analysis contains a well-developed theory of multidimensional Hilbert transforms with explicit convolution kernels for particular multiplier choices (Brackx, De Knock, and De Schepper).

Sangston's framework subsumes these constructions as follows. Let $\mathbb{G}_L$ denote the real geometric (Clifford) algebra generated by basis vectors $e_1,\dots,e_L$ with $e_k^2=1$ and $e_je_k=-e_ke_j$ for $j\ne k$; its elements are multivectors, sums of scalar, vector, bivector, and higher-grade parts. For $L=4n+3>N$ the unit pseudoscalar $I=e_1\cdots e_L$ is central and satisfies $I^2=-1$, so it can replace the imaginary unit in the Fourier transform of $\mathbb{G}_L$-valued functions on $\mathbb{R}^N$. One chooses an idempotent-valued function $\psi_{\mathcal{H}}(\boldsymbol{\omega})$ (with $\psi_{\mathcal{H}}^2=\psi_{\mathcal{H}}$) and defines the generalized analytic signal as the signal retaining the spectral components selected by $\psi_{\mathcal{H}}$: the source writes $\mathcal{F}[f_A]=\psi_{\mathcal{H}}\,\mathcal{F}[f]$, while the relation $f_A=f+f_{\mathcal{H}}$ introduced below gives $\mathcal{F}[f_A]=(1+a_{\mathcal{H}})\,\mathcal{F}[f]=2\psi_{\mathcal{H}}\,\mathcal{F}[f]$; this statement adopts the normalization $f_A=f+f_{\mathcal{H}}$ (the factor-of-2 discrepancy is inherited from the source and affects neither the idempotency condition, nor the operator questions below, nor the boundary-value status of $f_A$, which is unchanged under multiplication by a nonzero scalar). The associated multiplier $a_{\mathcal{H}}=2\psi_{\mathcal{H}}-1$ then satisfies $a_{\mathcal{H}}^2=1$ and defines the extended Hilbert operator $\mathcal{H}=\mathcal{F}^{-1}a_{\mathcal{H}}\mathcal{F}$, which formally squares to the identity and toggles between $f$ and its extended Hilbert transform $f_{\mathcal{H}}=\mathcal{H}[f]$, with $f_A=f+f_{\mathcal{H}}$. Suitable choices of $\psi_{\mathcal{H}}$ reproduce the single-orthant, hypercomplex, and monogenic signals, and further choices produce genuinely new signals, including generic analytic signals with a polar form $f_A=R\,e^{I\theta}$ carrying a vector-valued phase.

The framework is, by its author's statement, developed 'formally through examples rather than mathematical rigor'. Two gaps are explicit in the source. First, the paper imposes sufficient conditions on the multiplier — it has no scalar part, and its odd-grade and even-grade components are $a_{\mathcal{H},o}=\lambda_o\hat u_o$ and $a_{\mathcal{H},e}=\lambda_e\hat u_e$ with real $\lambda_o,\lambda_e$ satisfying $\lambda_o^2-\lambda_e^2=1$ and unit multivectors $\hat u_o,\hat u_e$ satisfying $\hat u_o^2=1$, $\hat u_e^2=-1$, $\hat u_o\hat u_e+\hat u_e\hat u_o=0$ — but 'we do not know if they are necessary'. Second, the extended Hilbert operator is defined only through the frequency-domain multiplier, and 'an appropriate (convolutional) operator need not exist for an arbitrary idempotent'; no conditions characterizing when a spatial convolution kernel exists are given. The one-dimensional analytic signal is the boundary value of a holomorphic function on the upper half-plane, and the classical multidimensional signals are likewise obtainable from suitably defined Riemann-Hilbert boundary-value problems, but nothing similar is known for the generalized analytic signals of this framework. Closing these gaps is the subject of the present problem.

Problem Statement

Establish the rigorous mathematical foundations of the idempotent-generated framework for multidimensional analytic signals of Sangston (2024). All four requested results below serve this single objective — characterizing the admissible idempotents and their associated transforms — for the framework as defined in the source (signals $f:\mathbb{R}^N\to\mathbb{R}$, algebra $\mathbb{G}_L$ with $L=4n+3>N$, Fourier transform with the central pseudoscalar $I$ in place of the imaginary unit, idempotent $\psi_{\mathcal{H}}$ and multiplier $a_{\mathcal{H}}=2\psi_{\mathcal{H}}-1$):

(a) Idempotency characterization: determine necessary and sufficient conditions on a multivector-valued Fourier multiplier $a_{\mathcal{H}}(\boldsymbol{\omega})$ in $\mathbb{G}_L$ (equivalently on the idempotent $\psi_{\mathcal{H}}=(1+a_{\mathcal{H}})/2$) for $a_{\mathcal{H}}^2=1$ pointwise in $\boldsymbol{\omega}$. In particular, decide whether the source's sufficient conditions — no scalar part, with odd-grade and even-grade components $\lambda_o\hat u_o$ and $\lambda_e\hat u_e$, $\lambda_o,\lambda_e$ real with $\lambda_o^2-\lambda_e^2=1$, and unit multivectors $\hat u_o,\hat u_e$ with $\hat u_o^2=1$, $\hat u_e^2=-1$, $\hat u_o\hat u_e+\hat u_e\hat u_o=0$ — are necessary, replacing or refining them if not.

(b) Operator existence and kernels: for a given idempotent $\psi_{\mathcal{H}}$, characterize the algebraic and analytic conditions under which the extended Hilbert operator $\mathcal{H}=\mathcal{F}^{-1}a_{\mathcal{H}}\mathcal{F}$ exists as a well-defined operator on an explicitly stated function space (for example $L^2(\mathbb{R}^N;\mathbb{G}_L)$ or the Schwartz class), and determine when $\mathcal{H}$ admits a representation as a spatial convolution operator, identifying the convolution kernel explicitly where it exists.

(c) Singular-integral reading: determine which multidimensional singular-integral operators of Calderon-Zygmund type arise as extended Hilbert operators, and characterize how the corresponding idempotents produce them, thereby connecting the framework to the established Calderon-Zygmund theory of the Riesz transforms and their Clifford generalizations. This part refines (b) and is not discharged by it: because a Calderon-Zygmund operator is a convolution operator with a homogeneous kernel, the convolution characterization required in (b) is presupposed, but the Calderon-Zygmund classification of the resulting kernels (homogeneity of degree $-N$ together with the standard size and smoothness conditions) and the identification of the arising operators with the classical Riesz-type transforms must still be established explicitly.

(d) Boundary-value structure: determine for which idempotents the generalized analytic signal $f_A=f+f_{\mathcal{H}}$ is the boundary value, on $\mathbb{R}^N=\partial\mathbb{R}^{N+1}_{+}$, of a function that is monogenic in the upper half-space $\mathbb{R}^{N+1}_{+}=\{(x_0,\boldsymbol{x}):x_0>0\}$ (with values in $\mathbb{G}_L$ and annihilated by $D=\partial_{x_0}+\sum_{k=1}^{N}e_k\partial_{x_k}$), or more generally of a function solving a Riemann-Hilbert-type boundary-value problem (that is, a boundary-value or transmission problem with prescribed Clifford-valued boundary data), and identify the precise boundary-value problem corresponding to a given idempotent choice.

References

1. K. James Sangston, 'A Geometric Algebra Framework for a Multidimensional Analytic Signal', arXiv preprint arXiv:2411.10412, 2024, DOI 10.48550/arxiv.2411.10412, https://arxiv.org/abs/2411.10412
2. T. Bülow and G. Sommer, 'Hypercomplex signals - a novel extension of the analytic signal to the multidimensional case', IEEE Transactions on Signal Processing 49(11), 2001, pp. 2844-2852, DOI 10.1109/78.960432, https://doi.org/10.1109/78.960432
3. M. Felsberg and G. Sommer, 'The monogenic signal', IEEE Transactions on Signal Processing 49(12), 2001, DOI 10.1109/78.969520, https://doi.org/10.1109/78.969520
4. S. L. Hahn, 'Multidimensional complex signals with single-orthant spectra', Proceedings of the IEEE 80(8), 1992, DOI 10.1109/5.158601, https://doi.org/10.1109/5.158601
5. F. Brackx, B. De Knock, and H. De Schepper, 'Generalized multidimensional Hilbert transforms in Clifford analysis', International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article 98145, 2006, DOI 10.1155/ijmms/2006/98145, https://doi.org/10.1155/ijmms/2006/98145
6. F. Brackx, B. De Knock, and H. De Schepper, 'Hilbert Transforms in Clifford Analysis', chapter in Geometric Algebra Computing (eds. E. Bayro-Corrochano and G. Scheuermann), Springer, 2010, DOI 10.1007/978-1-84996-108-0_9, https://doi.org/10.1007/978-1-84996-108-0_9
7. P. Cerejeiras and U. Kähler, 'Monogenic Signal Theory', chapter in Operator Theory (ed. D. Alpay), Springer Basel, 2014, pp. 1-22, DOI 10.1007/978-3-0348-0692-3_14-1, https://doi.org/10.1007/978-3-0348-0692-3_14-1
8. Mikhail Tsitsvero, Pierre Borgnat, and Paulo Gonçalves, 'Analytic signal in many dimensions', arXiv preprint arXiv:1712.09350, 2017, DOI 10.48550/arxiv.1712.09350, https://arxiv.org/abs/1712.09350
9. Brian Knight and Naoki Saito, 'A Structurally Coherent Spatial Phase Estimate', in Scale Space and Variational Methods in Computer Vision (SSVM 2025), Lecture Notes in Computer Science, Springer, 2025, pp. 311-323, DOI 10.1007/978-3-031-92369-2_24, https://doi.org/10.1007/978-3-031-92369-2_24