arXiv subjects

E. Palsson

Publications and source records attributed to E. Palsson.

5 recordsLinked to original sources

arXiv2026

PDE propagation, sampling, and the Fourier ratio

We study recovery from incomplete random spatial samples for discretized fields arising as fixed-time snapshots of partial differential equations. The organizing parameter is the Fourier ratio $$ FR(g)=\frac{\|\widehat g\|_1}{\|\widehat g\|_2}, $$ which quantifies effective spectral dimension and governs stable $\ell^1$ recovery in bounded orthonormal sampling models. Our main observation is that fixed-time PDE propagation can strictly improve Fourier ratio bounds relative to the discretized initial data. In dimension three, the wave snapshot operator introduces additional high-frequency decay, leading after discretization to Fourier ratio bounds that are uniformly controlled in the grid size (up to discretization errors), whereas the corresponding bounds for the initial discretization are typically polynomial in $N$. For the heat equation in any dimension, Gaussian frequency damping yields Fourier ratio bounds that are essentially independent of grid resolution for fixed positive time. Combining these deterministic Fourier ratio improvements with standard $\ell^1$ recovery guarantees yields explicit sampling-rate bounds for stable reconstruction from missing spatial samples. Numerical experiments confirm that PDE propagation acts as a spectral preconditioner that lowers effective sampling complexity in practice.

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arXiv2026

Discretization, sampling, and the Fourier ratio

We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in $C^{2}([0,1]^{2})$ sampled on an $N$ by $N$ grid, we show that a random subset of spatial samples of size $$ C\frac{r_{N}^{2}}{\eps^{2}}\log(r_{N}/\eps)^{2}\log(N^{2}) $$ suffices, with high probability, to recover the entire discretized signal via $\ell^{1}$ minimization with relative $L^{2}$ error $O(\eps)$. We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework.

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arXiv2025

The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure $\mu$ on $\mathbb{R}^d$ and $f\in L^2(\mu)$, the Fourier ratio compares $L^1$ and $L^2$ norms of a regularized Fourier transform at scale $R$. We develop a fractal uncertainty principle giving sharp two-sided bounds in terms of covering numbers of spatial and frequency supports, with applications to exact signal recovery. We show that small Fourier ratio implies efficient approximation by low-degree trigonometric polynomials in $L^1$, $L^2$, and $L^\infty$. In contrast, restriction estimates reveal a sharp gap between curved measures and random fractal measures, yielding strong lower bounds on approximation degree. Applications to convex surface measures are also obtained.

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arXiv2024

Multi-linear forms, structure of graphs and Lebesgue spaces

Consider the operator $$T_Kf(x)=\int_{{\mathbb R}^d} K(x,y) f(y) dy,$$ where $K$ is a locally integrable function or a measure. The purpose of this paper is to study the multi-linear form $$ Λ^K_G(f_1, \dots, f_n)=\int \dots \int \prod_{ \{(i,j): 1 \leq i<j \leq n; E(i,j)=1 \} } K(x^i,x^j) \prod_{i=1}^n f_i(x^i) dx^i, $$ where $G$ is a connected graph on $n$ vertices, $E$ is the edge map on $G$, i.e $E(i,j)=1$ if and only if the $i$'th and $j$'th vertices are connected by an edge, $K$ is the aforementioned kernel, and $f_i: {\mathbb R}^d \to {\mathbb R}$, measurable. This paper establishes multi-linear inequalities of the form $$ Λ^K_G(f_1,f_2, \dots,f_n) \leq C {||f_1||}_{L^{p_1}({\mathbb R}^d)} {||f_2||}_{L^{p_2}({\mathbb R}^d)} \dots {||f_n||}_{L^{p_n}({\mathbb R}^d)}$$ and determines how the exponents depend on the structure of the kernel $K$ and the graph $G$.

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arXiv2014

A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral

We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d) s_0(d)$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \cite{GI12} and in higher dimensions \cite{GGIP12}. In this paper, we improve upon those exponents, using a group-theoretic method that sheds new light on the classical approach to these problems. The key to our approach is a group action perspective which leads to natural and effective formulae related to the classical Mattila integral.

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