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Uniqueness and non-uniqueness pairs for the fractional Laplacian

This paper establishes sufficient conditions on discrete subsets of $\mathbb{R}^d$ to determine whether they form uniqueness or non-uniqueness pairs for the fractional Laplacian, while also extending these results to a broader class of multiplier operators.

Original authors: Ricardo Motta

Published 2026-04-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ricardo Motta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but the clues you have are very strange. You are looking for a hidden object (let's call it a "function" or a "signal") that exists everywhere in space.

Usually, to find something, you need to see it clearly. But in this mathematical world, the object is invisible unless you look at it from a very specific angle or check specific spots.

This paper, written by Ricardo Motta, is about a game of "Find the Hidden Signal" played with a special mathematical tool called the Fractional Laplacian.

Here is the breakdown of the game, the rules, and the surprising twists, explained without the heavy math jargon.

The Game Setup: Two Types of Clues

Imagine you have a mysterious signal, $f$ . You don't know what it looks like, but you have two ways to investigate it:

The "Where is it?" Check (Set $\Lambda$ ): You check a specific list of locations (a discrete set of points) to see if the signal is zero there. If the signal is zero at these spots, it's like finding footprints that stop at a certain point.
The "How is it moving?" Check (Set $M$ ): You use the Fractional Laplacian (let's call it the "Magic Scanner") to check how the signal behaves at a different list of locations. This scanner measures the "curvature" or "spread" of the signal in a non-local way (meaning it looks at the whole picture, not just the immediate neighborhood).

The Big Question: If you find that the signal is zero at the first list of spots ( $\Lambda$ ) AND the Magic Scanner reads zero at the second list of spots ( $M$ ), does that mean the signal is completely zero everywhere? Or could there be a hidden, non-zero signal that just happens to look zero at those specific spots?

The Two Outcomes

The paper investigates when the answer is "Yes, it must be zero" versus "No, a hidden signal could exist."

1. The "Uniqueness" Case (The Signal is Caught)

Sometimes, the lists of spots you check are so dense or arranged in a specific pattern that they trap the signal.

The Analogy: Imagine trying to hide a whisper in a crowded room. If you check every single person's ear (a very dense list), you will definitely hear the whisper if it exists.
The Result: Motta proves that if your lists of spots ( $\Lambda$ and $M$ ) are "dense enough" (mathematically speaking, they grow at a certain rate as you go further out), then the only signal that fits the clues is the empty signal (zero everywhere). The signal is forced to vanish.

2. The "Non-Uniqueness" Case (The Signal Hides)

This is where it gets interesting. Sometimes, the lists of spots are too sparse or arranged in a way that allows a "ghost" signal to slip through.

The Analogy: Imagine a game of "Marco Polo" in a giant, empty field. If you only shout "Marco" at a few specific trees far apart, a swimmer (the signal) could be hiding in the water between the trees, completely undetected.
The Result: Motta shows that if the lists are "sparse enough" (they grow very slowly or are far apart), you can construct a clever, non-zero signal that looks perfectly zero at all your check-points. The signal is invisible to your specific test.

The Secret Ingredient: How Fast Do the Spots Grow?

The most important discovery in the paper is about how the spots are spaced out.

The "Slow Growers" (Logarithmic): If your list of spots grows very slowly (like the numbers $1, 2, 3...$ but spaced out like $\log(n)$ ), the signal cannot hide. Even though the spots are far apart, they are "dense enough" in a mathematical sense to catch the signal.
The "Fast Growers" (Power Laws): If your list of spots grows quickly (like $n^2$ or $n^{0.9}$ ), the signal can hide. There is too much empty space between the clues for the signal to escape detection.

The "Magic Scanner" vs. The "Fourier Transform"

The paper compares this "Fractional Laplacian" scanner to a famous tool called the Fourier Transform (which is used to break down sounds into frequencies).

The Fourier Transform is very strict. If you have a signal that is zero at certain spots, it's very hard to make it "disappear" from the scanner unless the signal is truly zero.
The Fractional Laplacian is more "loose" or "non-local." It allows for more wiggle room. Motta shows that for the Fractional Laplacian, you can have signals that are zero at a set of points and their "scanner reading" is zero at another set of points, yet the signal is still alive and kicking.

The "Ghost" Construction (How to Hide)

In the second half of the paper, Motta actually builds these "ghost" signals.

The Trick: He uses a mathematical technique involving "Blaschke products" (which are like building blocks for functions with specific zeros).
The Analogy: Think of it like building a house of cards. He carefully arranges the cards (the mathematical functions) so that they collapse to zero exactly where you look (at $\Lambda$ and $M$ ), but they stand tall and strong everywhere else.
The Surprise: He proves that for the Fractional Laplacian, you can hide a signal even if the "hiding spots" are arranged in a very regular grid (like a lattice). This is a major difference from other mathematical tools where a regular grid would catch the signal immediately.

Why Does This Matter?

This isn't just about abstract math; it's about control and observation.

Physics: The Fractional Laplacian models things like heat diffusion in complex materials or how particles jump in a random walk (Lévy flights). Knowing when you can uniquely identify a state based on sparse measurements is crucial for engineering and physics.
The "Uncertainty Principle": This work touches on the idea that you can't know everything about a system at once. If you try to pin down a signal too tightly in space (by checking too many points), you might lose information about its behavior elsewhere. Motta shows exactly where that line is drawn.

Summary in One Sentence

Ricardo Motta's paper maps out the exact rules of a game where you try to catch a hidden mathematical signal using two different types of clues; he discovers that if your clues are spaced out just right, the signal is forced to vanish, but if they are spaced too far apart, a "ghost" signal can hide in plain sight, undetected by your tools.

1. Problem Statement

The paper addresses Question A in the context of the fractional Laplacian operator $(-\Delta)^s$ ( $0 < s < 1$ ) acting on discrete subsets of $\mathbb{R}^d$ . Specifically, given two discrete sets $\Lambda, M \subset \mathbb{R}^d$ and a function space $H$ , the author investigates under what conditions the following system implies that a function $f \in H$ must vanish identically:
$\begin{cases} f(x) = 0 & \text{for all } x \in \Lambda, \\ (-\Delta)^s f(x) = 0 & \text{for all } x \in M. \end{cases}$

If the only solution is $f \equiv 0$ , the pair $(\Lambda, M)$ is called a Uniqueness Pair (UP).
If there exists a non-zero solution, it is called a Non-Uniqueness Pair (NUP).

The study is motivated by recent developments in Fourier analysis (e.g., Radchenko-Viazovska interpolation formulas) and the theory of unique continuation for partial differential equations. The central challenge is determining how the density and asymptotic distribution of discrete sampling sets influence the rigidity of solutions to non-local equations.

2. Methodology

The author employs a combination of harmonic analysis, complex analysis, and geometric measure theory. The methodology is divided into two main approaches corresponding to the two types of results:

A. Uniqueness Proofs (The "Density" Approach)

To prove uniqueness, the paper utilizes:

Decay Estimates: The author establishes that if a function vanishes on a sufficiently dense discrete set $\Lambda$ , the function itself (and its fractional Laplacian) must exhibit specific exponential or polynomial decay at infinity. This relies on the lower Beurling density of the set under a specific change of variables.
Analytic Continuation: By combining the decay of $f$ and $(-\Delta)^s f$ with the definition of the fractional Laplacian via the Fourier transform ( $\widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat{f}(\xi)$ ), the author shows that the Fourier transform $\hat{f}$ extends to a holomorphic function in a complex tube.
Zero Sets of Holomorphic Functions: Using the fact that $|\xi|^{2s}$ is not smooth at the origin (for non-integer $s$ ), the author proves that $\hat{f}$ must have a zero of infinite order at the origin. Combined with analyticity, this forces $\hat{f} \equiv 0$ .
De Branges Spaces: For the Hilbert transform and specific 1D cases, the author leverages the theory of de Branges spaces and model spaces to relate zero sets to interpolation properties.

B. Non-Uniqueness Proofs (The "Construction" Approach)

To prove non-uniqueness, the author constructs explicit counterexamples:

Gelfand-Shilov Spaces: The construction takes place in highly regular spaces $S^{(p,q)}_s(\mathbb{R}^d)$ (subspaces of Schwartz space with specific exponential decay).
Blaschke Products and Inner Functions: In dimension $d=1$ , the author constructs functions in the model space $K^2_\Theta$ associated with a meromorphic inner function $\Theta$ . By carefully choosing the zeros of the Blaschke product to match the target sets $\Lambda$ and $M$ , and ensuring the phase function satisfies specific density conditions, a non-zero function is constructed that vanishes on both sets.
Lattice Periodization: For higher dimensions ( $d \ge 1$ ), the author exploits the structure of lattices. By defining the Fourier transform $\hat{f}$ to be supported on a union of shifted fundamental domains of a dual lattice, the problem reduces to solving a linear system of equations for the values of $\hat{f}$ on these shifts. If the system has a non-trivial solution, a non-zero $f$ exists.

3. Key Contributions and Results

A. Uniqueness Results (Theorems 1.1 & 1.3)

The paper provides sufficient conditions on the lower density of sets $\Lambda$ and $M$ to guarantee uniqueness.

Geometric Setup: The density is measured using a change of variables $\Phi_{\alpha, c}$ or $G_\alpha$ . For example, $D^-_{\Phi_{1,c}}(\Lambda) > 0$ implies the set is "dense enough" relative to an exponential scale.
Main Theorem 1.1:
- If $d \ge 1$ and both sets have positive lower density under the map $\Phi_{1,c}$ (linear scale), uniqueness holds.
- If $d=1$ , mixed conditions work: e.g., $\Lambda$ dense under linear scaling and $M$ dense under power scaling ( $G_\alpha$ ) implies uniqueness.
Corollary: This confirms that sets like $\{ \pm c \log n \}$ (slow growth) combined with $\{ \pm c n^\alpha \}$ (polynomial growth) form uniqueness pairs for the fractional Laplacian.

B. Non-Uniqueness Results (Theorems 1.2 & 1.4)

The paper demonstrates that uniqueness is not guaranteed for all discrete sets, even in highly regular function spaces.

Theorem 1.2 (1D Case): If the upper density of sets $\Lambda$ $Λ$ and $M$ $M$ (under the map $G_\alpha$ $G_{α}$ ) is finite, then $(\Lambda, M)$ $(Λ, M)$ is a Non-Uniqueness Pair for the half-Laplacian ( $s=1/2$ $s = 1/2$ ) in Gelfand-Shilov spaces.
- Significance: This contrasts sharply with the Fourier transform, where similar density conditions often imply uniqueness. It highlights the "stronger non-locality" of the fractional Laplacian compared to the standard Fourier transform.
Theorem 1.4 (Higher Dimensions): For any invertible matrix $A$ $A$ , the pair of lattices $(A\mathbb{Z}^d, A\mathbb{Z}^d)$ $(A Z^{d}, A Z^{d})$ is a Non-Uniqueness Pair for the fractional Laplacian in the Schwartz space.
- Mechanism: The proof constructs a function whose Fourier transform is supported on three specific shifts of a fundamental domain, satisfying the vanishing conditions via a linear algebra argument.

C. Generalizations

The results are extended to a broader class of multiplier operators $T_m$ (Theorems 6.1 and 6.2), covering shifted fractional Laplacians, wave/Schrödinger propagators, and mixed multipliers.

4. Significance and Impact

Discrete Unique Continuation: The paper establishes a discrete analogue of unique continuation principles. It shows that for the fractional Laplacian, the "measure" of the set is less important than its asymptotic growth rate. Sparse sets that grow slowly (like logarithmic) enforce uniqueness, while sets growing polynomially (like $n^\alpha$ ) allow for non-trivial solutions.
Comparison with Fourier Transform: A major finding is the divergence between Fourier uniqueness and fractional Laplacian uniqueness. While the Fourier transform has strong rigidity for certain discrete sets (e.g., Radchenko-Viazovska), the fractional Laplacian admits non-uniqueness for sets that would be unique for the Fourier transform. This suggests the fractional Laplacian has a more complex non-local structure.
Boundary Value Interpretation: The results are interpreted via the Caffarelli-Silvestre extension. The vanishing of $f$ and $(-\Delta)^s f$ on discrete sets corresponds to the vanishing of the Dirichlet and weighted Neumann data on the boundary of a degenerate elliptic equation in the upper half-space. The paper proves that discrete boundary data can determine the solution uniquely or not, depending on the density.
New Function Spaces: The introduction and utilization of Gelfand-Shilov spaces $S^{(p,q)}_s$ provide a refined framework for studying non-uniqueness, showing that non-uniqueness persists even in spaces of functions with rapid decay and smoothness.

In summary, Motta's work provides a comprehensive characterization of when discrete sampling of a function and its fractional Laplacian determines the function uniquely, bridging the gap between harmonic analysis, PDE theory, and complex analysis.