Basis & Complete Massive Kernels

• Basis and complete massive kernels are families of kernel functions that span entire function spaces, ensuring optimal representation and reconstruction.
• They are constructed using analytic derivatives, group-theoretic methods, and parameter expansions to meet rigorous completeness criteria.
• These kernels have practical applications in RKHS theory, adaptive signal decomposition, equivariant machine learning, and quantum state reconstruction.

Basis and complete massive kernels encompass a diverse set of mathematical structures central to signal decomposition, equivariant machine learning, functional analysis, and quantum physics. The notion of completeness refers to whether a family of kernel functions or basis elements spans the entire relevant function space, allowing for optimal representation or reconstruction. In both classical and modern research, completeness criteria and construction techniques for kernel systems have deep implications for algorithmic efficiency, convergence rates, and physical interpretability.

1. Kernel Systems and Completeness in Functional Analysis

Kernel families, particularly in Hilbert spaces of analytic functions, play a fundamental role in reproducing kernel Hilbert spaces (RKHS). On the unit disc $D = \{z \in \mathbb{C}: |z| < 1\}$, the Hardy space $H^2(D)$ is equipped with the Szegő kernel:

$$K(z, w) = \frac{1}{1 - z\overline{w}}, \quad z, w \in D$$

which possesses the reproducing property $f(w) = \langle f, K(\cdot, w) \rangle$ for every $f \in H^2(D)$ (Lin et al., 2023). Classical bases such as $\{z^n\}$ (Fourier basis) and Blaschke products are incomplete for representing higher-frequency or singular components. To achieve completeness, the Szegő kernel dictionary is extended by taking all anti-holomorphic derivatives:

$$k_{n,w}(z) := \left( \frac{\partial}{\partial \overline{w}} \right)^n K(z, w)$$

yielding explicit formulas and normalized elements $e_{n,w}(z)$ that span a complete dictionary $$\mathcal{D}_{\text{complete}} = \{e_{n,w} : n \in \mathbb{N}_0, w \in D\}$$. Completeness is certified via the boundary-vanishing condition (BVC): $$\lim_{|w| \to 1 \text{ or } n \to \infty} \langle f, e_{n,w} \rangle = 0$$ for $f \in H^2(D)$, ensuring the existence of best $n$-term approximants (Lin et al., 2023).

2. Mean-Frequency Decomposition and Algorithmic Applications

Frequency analysis using complete massive kernels leverages the mean-frequency concept:

$$\mathrm{MF}(f) := \int_0^{2\pi} \theta'_\phi(t)\,dt + \int_0^{2\pi} \theta'_s(t)\,dt$$

where $f(z)$ factorizes into Blaschke, singular inner, and outer functions, and the indices $\theta'_\phi, \theta'_s$ quantify frequency content. The complete Szegő dictionary enables representations capable of resolving all frequency levels, outperforming classical kernel sets. Sparse decomposition algorithms—greedy (GA), orthogonal greedy (OGA), adaptive Fourier decomposition (AFD), pre-orthogonal AFD (POAFD), unwinding Blaschke expansions, and $n$-Best selection—are analyzed for convergence, with $n$-Best yielding the strongest theoretical guarantee. For signals $f \in H^2_\sigma$ (Hardy-Sobolev space of order $\sigma > 0$), convergence rates of $O(n^{-\sigma})$ match those of Fourier and Laguerre expansions (Lin et al., 2023).

3. Body-Ordered and Equivariant Complete Kernels in Geometric Learning

In geometric machine learning, the construction of complete, body-ordered, equivariant kernels is exemplified by Wigner kernels. These kernels, especially in atomic and molecular ML, capture all $(\nu+1)$-body correlations without explicit feature-space truncation. The recursive Wigner-iteration formula uses Clebsch-Gordan coefficients to project iterated neighbor densities $\rho_i(\mathbf{x})$ onto irreducible representations:

$$k^{(\nu+1),\lambda}_{\mu\mu'}(A_i,A'_{i'}) = \sum C^{\ell_1\ell_2\lambda}_{m_1 m_2\,\mu} k^{\nu,\ell_1}_{m_1 m_1'} k^{1,\ell_2}_{m_2 m_2'} C^{\ell_1\ell_2\lambda}_{m_1' m_2'\,\mu'}$$

where $A_i$ are atomic environments and $\lambda$ indexes irreducible components. The completeness is established by identifying $k^{\nu,\lambda}$ with scalar products in an untruncated $\nu$-body feature space. Computational complexity scales as $O(\nu_{\text{max}}\lambda_{\text{max}}^7)$, independent of radial or chemical basis size. Empirical results demonstrate state-of-the-art accuracy on QM9 atomization energy and dipole benchmarks (Bigi et al., 2023).

4. Model Spaces, Riesz Bases, and Completeness via Schur–Nevanlinna Parameters

In the theory of model spaces $K_\theta = H^2 \ominus \theta H^2$ (with inner $\theta$), reproducing kernel systems $k^\theta_\lambda$ are organized as Riesz sequences or bases, with completeness characterized by Schur–Nevanlinna (SN) parameters $\{\gamma_n\}$. The SN iteration constructs inner functions via prescribed parameter sets, where the summability $\sum |\gamma_n| < \infty$ guides completeness. For Carleson/Blaschke sequences $\Lambda$, depending on the SN parameters and limiting behavior of $\theta(\lambda_n)$, one can build either incomplete Riesz sequences or fully complete Riesz bases. Compactness criteria for Hankel operators $H_{\theta \overline{B}}$ link the vanishing condition $\lim_{n \to \infty} \theta(\lambda_n) = 0$ directly to completeness of the kernel system (Boricheva, 2022).

5. Quantum Complete Bases and Massive Kernels from Canonical Quantization

Canonical quantization techniques enable the systematic construction of complete basis sets via point transformations $x \mapsto W(x)$ and conjugate momentum $p_W$ in quantum systems. A parametric family of momentum operators $\hat{p}_W^{(a)}$—quasi-Hermitian in $x$, Hermitian in $W$—yields four classes of complete bases:

• Continuous mutually unbiased bases (MUB)
• Orthogonal bases ($\delta$-normalized)
• Biorthogonal bases ($\delta$-normalized)
• W-harmonic oscillator bases (Hermite functions) and coherent states

Each basis is associated with an explicit massive kernel:

$$K(x,p_W) = \frac{[J(x)]^\gamma}{\sqrt{2\pi\hbar}} \exp\left[\frac{i}{\hbar} \frac{p}{m} W(x)\right]$$

where $J(x) = dW/dx$ and the choice of $\gamma$ depends on operator ordering. Completeness and orthonormality are ensured by careful treatment of Jacobian and mass-scaling factors. The spectrum mapping $p_W \to p/(mi)$ is permitted, preserving completeness (Kouri et al., 2016).

6. Steerable Equivariant Kernels and Completeness in Poincaré-Group Convolutions

In equivariant convolutional neural networks, completeness of the kernel basis under pseudo-Euclidean (including Minkowski) groups is essential for expressivity, particularly in the context of massive representations (e.g., spinor fields under the Poincaré group). Classical Clifford-Steerable CNNs (CSCNNs) often produce incomplete bases, missing higher-order Clebsch-Gordan channels unless stacked further. Augmenting the kernel space with auxiliary, translation-invariant multivectors from the input feature field enables construction of a provably complete set:

$$K(z, \zeta) = \sum_\ell \phi_\ell(r) I_\ell(z, \zeta)$$

where $\ell$ runs over allowed grades, $\phi_\ell(r)$ are radial profiles, and $I_\ell$ are group-theoretic intertwiners. For Poincaré representations, this construction recovers all Dirac bilinear covariants (scalar, vector, tensor, axial, pseudo-vector) in a single layer, matching the full harmonic steerable kernel space and yielding maximal expressivity for PDE modeling (Szarvas et al., 15 Oct 2025).

7. Synthesis and Cross-Disciplinary Context

Complete massive kernel bases furnish optimal representational efficiency across functional, quantum, and geometric domains. Their construction often involves parameter derivatives, group-theoretic recursion, spectral theory, and compactness arguments. Applicability ranges from sparse signal decomposition using adaptive algorithms (Lin et al., 2023), high-fidelity atomistic machine learning (Bigi et al., 2023), and quantum state reconstruction (Kouri et al., 2016), to the architectural design of expressive equivariant neural networks (Szarvas et al., 15 Oct 2025).

A plausible implication is that completeness, when rigorously realized via analytic, algebraic, or computational criteria, governs the practical efficacy of kernel-based methods across signal processing, ML, and mathematical physics. Advances in kernel basis construction—such as parameter-derivative dictionaries, Wigner recursions, and conditional augmentation—systematically resolve limitations in expressivity, convergence, and physical fidelity.