Square-Integrable Kernels Overview

• Square-integrable interaction kernels are defined as kernels in L² that enable Hilbert–Schmidt operator analysis, making them essential for spectral theory and diverse applications.
• They are constructed via direct integrals, mixtures, and Cauchy-type representations, forming robust frameworks such as reproducing kernel Hilbert spaces for efficient analysis.
• Their applications span operator theory, stochastic systems, inverse problems, and representation theory, ensuring stability and well-posedness in both deterministic and probabilistic settings.

Square-integrable interaction kernels are central objects in mathematical analysis, probability theory, operator theory, representation theory, and mathematical physics. They arise whenever one studies integral operators or stochastic interactions whose associated kernels are in $L^2$ (or are locally square-integrable in a suitable sense), thereby allowing Hilbert space methods, spectral theory, and related analytical techniques to be applied. This comprehensive overview elucidates the definition, analytical properties, construction methods, key applications, and representative results concerning square-integrable interaction kernels across major areas of current research.

1. Definition and Analytical Foundations

A kernel $K(x,y)$ defined on a measurable space $X \times Y$ (with suitable measures) is called square-integrable if $K \in L^2(X \times Y)$, i.e.,

$\iint_{X \times Y} |K(x,y)|^2 \, dx\, dy < \infty.$

Such kernels are the integral kernels of Hilbert–Schmidt operators when acting on $L^2$ spaces; the Hilbert–Schmidt property is a critical threshold for compactness and spectral theory.

In many settings, the square-integrability may hold only locally (for example, when integrating over compact subsets as in "locally of trace class" kernels (Bufetov et al., 2017)). In system identification and stochastic process theory, a more refined property, diagonally square root integrability (DSRI), is required: $\int \sqrt{K(t, t)} dt < \infty$ or $\sum_{t} \sqrt{K(t, t)} < \infty$ for discrete time (Khosravi et al., 2023). This condition guarantees stability and further ensures the boundedness of operators defined by the kernel.

Square-integrable kernels naturally endow the associated Hilbert space of functions with an inner product, often forming a reproducing kernel Hilbert space (RKHS) (Hotz et al., 2012, Bufetov et al., 2017, Khosravi et al., 2023), where evaluation at a point is a continuous linear functional. Such spaces afford powerful tools from complex and functional analysis.

2. Construction and Structure of Square-Integrable Kernels

Direct Integrals and Mixtures

Many important square-integrable kernels are constructed as integrals or "scale mixtures" of elementary kernels. For example, given a family $\left\{K_\omega\right\}_{\omega \in \Omega}$ of reproducing kernels, the integral kernel

$$K(x, y) = \int_{\Omega} K_{\omega}(x, y)\, d\mu(\omega)$$

is square-integrable if $\int K_{\omega}(x, x) d\mu(\omega) < \infty$ for each $x$ (Hotz et al., 2012). Classical Mercer kernels, kernels from integral transforms (Fourier, Laplace), and scale-mixtures of RBFs (radial basis functions) fall in this framework.

Division Properties and Integrable Forms

In spaces where a "division property" holds—i.e., if $f(p) = 0$ then $f(x)/(x-p)$ remains in the space—one often finds Cauchy-type integrable representations:

$K(x, y) = \frac{A(x)B(y) - B(x)A(y)}{x - y}$

with $A, B$ constructed from the kernel itself (Bufetov et al., 2017). This structure underpins a large class of integrable systems, including the sine kernel, Airy kernel, and Bessel kernel from random matrix theory.

Operator-valued and Non-symmetric Kernels

Recent developments allow for square-integrable kernels without symmetry assumptions, especially in nonlocal calculus and peridynamics (Foss et al., 27 Aug 2024). Compactness, integration by parts, and nonlocal-to-local convergence properties are studied with minimal symmetry, provided square-integrability and suitable support conditions for the kernel.

3. Square-integrable Kernels in Operator and Spectral Theory

Square-integrable kernels provide the foundation for the Hilbert–Schmidt class of integral operators:

$(Tf)(x) = \int_{Y} K(x, y) f(y) \, dy$

with $\|T\|_{HS}^2 = \|K\|_{L^2(X \times Y)}^2$. Such operators are compact, possess spectral decompositions, and in the self-adjoint case enjoy trace-class properties under further conditions. Local square-integrability (Hilbert–Schmidt on compacts) is required for spectral analysis on noncompact or singular spaces (Bufetov et al., 2017).

Explicit formulas for square-integrable kernels are central to scattering theory and PDE analysis, as in the case of the resolvent and heat kernels for the Schrödinger operator with inverse square potential, expressed via Bessel and hypergeometric functions (Moustapha, 2017). Furthermore, generalized notions of integrals extend square-integrable kernel formulas to regimes where standard integrals diverge, especially when regularizing Green functions in higher dimensions or for point interactions (Dereziński et al., 2023).

Integration by parts and compactness properties are established for nonlocal operators with integrable, possibly non-symmetric, kernels, and nonlocal-to-local convergence (approximating derivatives in the limit) can be rigorously proved under square-integrability assumptions (Foss et al., 27 Aug 2024).

4. Applications in Probability, Statistical Mechanics, and System Identification

Stochastic Systems and Regularization

In statistical mechanics and stochastic interacting particle systems, square-integrability of the "interaction kernel" is crucial for well-posedness of the (operator) perturbation series, guaranteeing convergence and controlling the evolution of the system, e.g., through Kato or Khasminskii-type conditions (Bogdan et al., 2012, Thiery et al., 2015).

In system identification, the DSRI condition ensures that Gaussian process priors placed on impulse responses yield almost surely stable (BIBO) systems; DSRI is both necessary and sufficient for the $L^1$-stability (Lemma 13 and Theorem 15 in (Khosravi et al., 2023)). Moreover, the RKHSs associated with DSRI kernels inherit desirable continuity properties for convolution and Fourier operators, a foundation for robust computational methods.

Inverse and Learning Problems

Square-integrable kernels play a central role in the theoretical analysis of inverse problems: the closure of the RKHS associated with the squared kernel (induced by the second derivative of a loss functional) defines the identifiable component of the interaction kernel in mean-field equations of interacting particles (Lang et al., 2021). Regularization (e.g., Tikhonov, SVD truncation) is necessary, as the inverse problem is ill-posed in infinite dimensions, but the square-integrability structure—particularly when using data-adaptive weighted $L^2$ spaces—provides both theoretical and practical guidance for kernel estimation.

5. Representation Theory and Harmonic Analysis

Square-integrable kernels arise naturally in representation theory, particularly as reproducing kernels in spaces of square-integrable representations (discrete series) of semisimple Lie groups (Vargas, 26 Mar 2025). The decomposition (branching laws) of a representation $V$ upon restriction to a subgroup $H$ is governed by integral operators whose kernels are square-integrable on $G$ (or $H \times G$):

$V \simeq \int_{\hat{H}} V_i \otimes M_i \, d\mu(i)$

where projection onto each piece and intertwining maps are realized as integral (or differential) operators with square-integrable kernels. These kernels mediate the interaction between different symmetry sectors and are crucial in harmonic analysis and in modeling quantum systems with symmetries.

The importance of square-integrable representations is also reflected in the context of translation-invariant systems (e.g., in time–frequency analysis via affine, Heisenberg, and shearlet groups (Linnell et al., 2015)), where the admissibility and completeness of bases are intimately tied to the square-integrability of the underlying kernel coefficients.

6. Variational Principles, Energy Problems, and Nonlocal PDEs

In potential theory and approximation, integrable (in particular, square-integrable) kernels are employed in the paper of extremal configurations for minimum energy and polarization problems (Simanek, 2015). The minimizers of discrete and continuous energy associated to a kernel $K$ converge to the equilibrium measure when $K$ satisfies suitable integrability and regularity properties. This connection links the asymptotic distribution of optimal configurations to the analytic structure of $K$.

In nonlocal PDEs and mechanics, square-integrable kernels underpin the definition of nonlocal operators (e.g., nonlocal Laplacians or gradients), which, under integrability and compact support assumptions, allow for the development of discretization schemes (e.g., hybrid discontinuous Galerkin methods (Du et al., 2019)). The square-integrability of the kernel is essential for well-posedness, convergence analysis, and for ensuring compatibility with the limiting local PDE as the nonlocality vanishes.

However, a caveat is noted: for certain classes of mean-free (antisymmetric) kernels, the resulting nonlocal derivative operator is compact, precluding the validity of Poincaré-type inequalities on infinite-dimensional spaces, thus limiting coercivity and leading to the presence of zero-energy modes (Foss et al., 27 Aug 2024).

7. Recent Developments and Open Directions

Recent research extends square-integrable kernel theory to more general contexts:

• Berezin–Toeplitz quantization and random holomorphic sections: Square-integrable random holomorphic sections constructed via Toeplitz kernels (with possibly non-smooth symbols) yield probabilistic models whose zeros equidistribute according to curvature and satisfy central limit theorems. Kernel asymptotics underpin large deviations and universality results in geometric quantization (Drewitz et al., 24 Apr 2024).
• Generalized integrals and renormalization: The notion of generalized integrals allows for analytic continuation and regularization of bilinear forms and Green functions associated with singular interaction kernels in higher dimensions, facilitating an operator-theoretic extension to otherwise divergent cases (Dereziński et al., 2023).
• Nonlocal-to-local convergence and operator compactness: The paper of operator-theoretic properties for nonlocal differential operators with compactly-supported, nonsymmetric integrable kernels, including new integration by parts formulas and compactness results, is an area of active development (Foss et al., 27 Aug 2024).

---

The theory of square-integrable interaction kernels hence forms a unifying thread across complex analysis, operator theory, stochastic processes, mathematical physics, and computational mathematics. Their analytical tractability and structural richness enable both concrete applications and broad generalizations, while recent advances continue to broaden the class of kernels and operators for which square-integrability yields robust mathematical structure.