Related papers: Pure point diffraction and Poisson summation

This article studies Paley's theory for lacunary Fourier series on (nonabelian) discrete groups. The results unify and generalize the work of Rudin for abelian discrete groups and the work of Lust-Piquard and Pisier for operator valued…

In this work we consider translation-bounded measures over a locally compact Abelian group $\mathbb{G}$, with particular interest for their so-called diffraction. Given such a measure $\Lambda$, its diffraction $\widehat{\gamma}$ is another…

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + \sigma(X_t) dW_t,$ with discretely sampled observations $X_0, X_\Delta, \dots, X_{n\Delta}$. We analyse the nonlinear inverse problem…

In this article, we derive a theoretical formalism that unifies the rigorous coupled wave analysis and the dynamical diffraction theory. Based on this formalism, we design a computational approach for the diffraction calculation for the…

We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic…

Diffusion models are typically trained using pointwise reconstruction objectives that are agnostic to the spectral and multi-scale structure of natural signals. We propose a loss-level spectral regularization framework that augments…

Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they…

We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane. These formulas simultaneously generalize the classical Poisson formula and Newton…

Diffusion models have emerged as powerful tools for solving inverse problems, yet prior work has primarily focused on observations with Gaussian measurement noise, restricting their use in real-world scenarios. This limitation persists due…

We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse…

We show that uniform approximate lattices in nilpotent Lie groups are subsets of model sets. This extends a theorem due to Yves Meyer about quasicrystals in Euclidean spaces. To do so we study relatively dense subsets of simply connected…

We prove a uniform effective density theorem as well as an effective counting result for a generic system comprising a polynomial with a mild homogeneous condition and several linear forms using Roger's second moment formula for the Siegel…

We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the…

We consider temperate distributions on Euclidean spaces with uniformly discrete support and locally finite spectrum. We find conditions on coefficients of distributions under which they are finite sum of derivatives of generalized lattice…

The optical reflection coefficient of a dielectric medium moving uniformly in the plane spanned by its surface is rigorously calculated using classical electrodynamics and special relativity, and expressed in the Fourier domain, as a…

We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of…

Given a finite subset $F$ of integer points in $\mathbb Z^d$, it is of interest to seek conditions on $F$ that allow it to multi-tile $\mathbb Z^d$ by translations. To this end, we give a discretized version of the Bombieri-Siegel formula,…

Discrete diffusion language models learn to reconstruct text from randomly masked inputs, yet under mild assumptions their denoiser already implements the exact Bayesian posterior over the original tokens. We prove that the expected…

We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our…

We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability…