Related papers: An elementary proof of the reconstruction theorem

Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with…

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which…

Feature attributions are a popular tool for explaining the behavior of Deep Neural Networks (DNNs), but have recently been shown to be vulnerable to attacks that produce divergent explanations for nearby inputs. This lack of robustness is…

In this paper, we show how one can view certain models in regularity structures as some form of geometric rough paths. This is performed by identifying the deformed Butcher-Connes-Kreimer Hopf algebra with a quotient of the shuffle Hopf…

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…

Let $\varphi_t : M \to M$ be a flow on a smooth closed connected manifold $M$ that preserves and expands a foliation $F$. We establish a theorem of propagation of regularity along the leaves of $F$ for sections of vector bundles satisfying…

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the…

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…

We prove a version of the implicit function theorem for Lipschitz mappings $f:\mathbb{R}^{n+m}\supset A \to X$ into arbitrary metric spaces. As long as the pull-back of the Hausdorff content $\mathcal{H}_{\infty}^n$ by $f$ has positive…

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space $X$ which are obtained by deleting singletons determine $X$…

The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and…

Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of…

This paper studies least-square regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of low-complexity. Indeed, they force…

We give a motivation and gentle introduction into the regularity structure and model introduced by Otto, Sauer, Smith and Weber, which fall into the framework of Hairer, but have a greedier index set than the one given by trees. We do this…

We investigate regularizations of distributional sections of vector bundles by means of nets of smooth sections that preserve the main regularity properties of the original distributions (singular support, wavefront set, Sobolev…

The reconstruction theorem deals with dynamical systems that are given by a map $T:X\to X$ of a compact metric space $X$ together with an observable $f:X \to \R$ from $X$ to the real line $\R$. In 1981, by use of Whitney's embedding…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial…

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with…

In the present paper, we use a generalised shift operator in order to define a generalised modulus of smoothness. By its means, we define generalised Lipschitz classes of functions, and we give their constructive characteristics.…

Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy.…