Fractional Discrete Riesz Kernel

• Fractional discrete Riesz kernel is a mathematical operator generalizing the classical discrete Laplacian to fractional orders, enabling nonlocal interactions with power-law decay.
• It utilizes explicit formulas and Fourier diagonalization to derive kernel representations on periodic lattices, ensuring consistency with continuum fractional Laplacians.
• Applications span anomalous diffusion, fractional quantum mechanics, and nonlocal elasticity, with efficient FFT-based algorithms supporting precise numerical simulations.

A fractional discrete Riesz kernel generalizes the classical discrete Laplacian and Riesz potential to fractional orders on discrete lattices, enabling nonlocal interactions and power-law decay in matrix representations and convolution formulas. This kernel is foundational for fractional discrete calculus, particularly in the analysis of anomalous diffusion, fractional quantum mechanics, and problems posed on finite lattices and periodic domains.

1. Mathematical Definition, Explicit Formulae, and Representations

On a one-dimensional periodic chain with $N$ sites, the fractional discrete Laplacian matrix is given by

$$\Delta_{\alpha,N}[p,q] = -\mu\,f^{(\alpha)}_N(|p-q|),$$

where $f^{(\alpha)}_N(\ell)$ encodes the fractional power-law coupling, and $\mu$ is the particle mass parameter. The characteristic function is

$$f^{(\alpha)}(\lambda) = \Omega_\alpha^2\,\lambda^{\alpha/2}, \quad \alpha > 0, \;\Omega_\alpha>0.$$

For the periodic $N$-ring, $f^{(\alpha)}_N(\ell)$ admits an image-sum representation

$$f^{(\alpha)}_N(\ell) = \sum_{n=-\infty}^{\infty} f^{(\alpha)}_\infty(\ell+nN),$$

with the infinite-chain coefficients

$$f^{(\alpha)}_\infty(r) = \Omega_\alpha^2\,(-1)^r\, \binom{\alpha}{\frac{\alpha}{2} + r},$$

where

$$\binom{\alpha}{\frac{\alpha}{2}+r} = \frac{\Gamma(\alpha+1)}{\Gamma(\frac{\alpha}{2}+r+1)\Gamma(\frac{\alpha}{2}-r+1)}.$$

Alternatively, via Fourier diagonalization,

$$f^{(\alpha)}_N(\ell) = \frac{\Omega_\alpha^2}{N} \sum_{l=0}^{N-1} e^{2\pi i\,l\,\ell/N}\,\Bigl[4\sin^2\frac{\pi l}{N}\Bigr]^{\alpha/2}.$$

The explicit kernel thus encapsulates long-range interactions between all pairs $(p,q)$, generalized from nearest-neighbor to fractional, algebraically decaying connections. For the infinite chain, the entries reduce to

$$\Delta_{\alpha,\infty}[p,q] = -\mu\,\Omega_\alpha^2\,(-1)^{|p-q|}\,\binom{\alpha}{\frac{\alpha}{2} + |p-q|}.$$

For non-integer $\alpha/2$, a gamma-function form is valid.

2. Periodic Continuum Limit and Connection to the Riesz Derivative

As the lattice constant $h \to 0$ with string length $L=Nh$ kept finite, scaling laws $\mu \sim h$ and $\Omega_\alpha^2 \sim h^{-\alpha}$ are imposed to preserve finite total mass and energy. Under this scaling, the discrete Laplacian converges to the continuum fractional Laplacian: $h^\alpha\,(-\frac{d^2}{dx^2})^{\alpha/2},$ with continuum elastic energy

$$V_\alpha \longrightarrow \frac{\rho_0 A_\alpha}{2} \int_0^L u^*(x) (-\frac{d^2}{dx^2})^{\alpha/2} u(x)\, dx.$$

The L-periodic fractional Laplacian kernel is

$$K^{(\alpha)}_L(x-x') = \frac{\alpha! \sin(\frac{\alpha \pi}{2})}{\pi} \sum_{n=-\infty}^{\infty} \frac{1}{|x-x'-nL|^{\alpha+1}}.$$

This is a periodized version of the Riesz kernel; for $L \to \infty$, only the $n=0$ term survives, recovering the classic Riesz singularity.

3. Discrete Riesz Potentials and Kernel Properties

The discrete fractional Riesz kernel is central in the theory of discrete convolution operators such as the Riesz potential $I_\alpha$ acting on sequences $f:\mathbb{Z}\rightarrow\mathbb{R}$: $$I_{\alpha} f(n) = \sum_{k \in \mathbb{Z} \setminus\{n\}} \frac{f(k)}{|n-k|^{1-\alpha}} = \sum_{k \in \mathbb{Z}} K_\alpha(n-k) f(k),$$ where

$$K_\alpha(m) = \begin{cases} |m|^{\alpha-1}, & m \ne 0, \ 0, & m = 0. \end{cases}$$

Key analytic features:

• Symmetry: $K_\alpha(-k) = K_\alpha(k)$
• Singularity: $K_\alpha(k) \to \infty$ as $k \to 0$ for $0 < \alpha < 1$
• Decay: $K_\alpha(k) \sim |k|^{\alpha-1}$ as $|k| \to \infty$ (not summable for $\alpha < 1$)
• Positivity: $K_\alpha(k) > 0$ for all $k \neq 0$ These properties reflect the nonlocal nature and power-law interaction fundamental to fractional models (Hao et al., 2023, Hu et al., 2024).

4. Continuum and Asymptotic Connections

In the continuum $\mathbb{R}$, the Riesz potential of order $\alpha$ is

$$I_\alpha^\text{cont} f(x) = \int_{\mathbb{R}} |x-y|^{\alpha-1} f(y)\,dy.$$

By replacing the integral with a sum and omitting normalization, the discrete Riesz kernel is a direct analog: $$I_{\alpha} f(n) = \sum_{k \in \mathbb{Z}} |k|^{\alpha-1} f(n-k).$$ In discrete settings, as $h \to 0$, the kernel's algebraic tail for large $|k|$ matches the continuum Riesz kernel: $K_s(k) \sim c_s |k|^{-(1+2s)} + O(|k|^{-(2+2s)}),$ where the scaling constant in one dimension is $$c_s = 4^s \Gamma(1/2 + s) / (\sqrt{\pi} |\Gamma(-s)|)$$, showing consistency between discrete and continuous fractional Laplacians (Ciaurri et al., 2015).

5. Weighted Inequalities and Functional Spaces

Discrete Riesz potentials, via the kernel $K_\alpha(k)$, induce convolution operators whose boundedness properties on discrete weighted Lebesgue and Morrey spaces are characterized by explicit criteria in terms of Muckenhoupt weights:

• For $0 < \alpha < 1$, $1 < p < 1/\alpha$, $1/q = 1/p - \alpha$:

$$\|I_\alpha x\|_{\ell^q(w^q)} \leq C \|x\|_{\ell^p(w^p)},$$

if and only if $w \in A(p,q)$. End-point and weak-type estimates are also established (Hu et al., 2024).

• Similar results hold for fractional maximal operators associated with the kernel.

This structure mirrors the classical continuous Muckenhoupt–Wheeden theory, with adaptation for the algebraic tail and singularity of the discrete kernel (Hao et al., 2023, Hu et al., 2024).

6. Computational Methods and Fast Algorithms

Fast and accurate evaluation of discrete fractional Laplacians and kernels is achieved using discrete convolution schemes. Efficient methods, such as those based on mapping $\mathbb{R}$ to a finite interval and using a modified midpoint quadrature, enable the kernel evaluation to be cast as circulant convolution and efficiently computed via FFT with $O(N \log N)$ complexity. The resulting discrete kernel matches the power-law decay at infinity, $w_k \sim |k|^{-1-2s}$, ensuring second-order accuracy in grid spacing (Cayama et al., 2022).

Numerical validation confirms error decay like $1/N^2$ for fixed refinement and convergence to the exact fractional Laplacian under grid refinement.

7. Physical Significance and Applications

Fractional discrete Riesz kernels give rise to nonlocal elasticity, anomalous diffusion, and fractional quantum mechanics. In discrete lattice models such as the finite periodic chain, these kernels encode long-range power-law interactions, generalizing classical elastic couplings. Their periodized versions enable modeling on finite domains with cyclic or string topology, critical for finite-size effects in physical systems.

Applications span anomalous transport (Levy flights), time and space-fractional evolution equations, nonlocal Schrödinger equations, and the rigorous development of discrete fractional calculus on lattices of arbitrary geometry (Michelitsch et al., 2014, Michelitsch et al., 2015). The representation of discrete Riesz kernels is crucial for both theoretical analysis and efficient numerical simulation in these domains.