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Concept

Espace de Schwartz

Publications associées (18)

Explicit representations for Banach subspaces of Lizorkin distributions

Michaël Unser, Sebastian Jonas Neumayer

The Lizorkin space is well suited to the study of operators like fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. In return, we show that it is dense in C0(Double- ...

2023

The inductive bias of deep learning: Connecting weights and functions

Guillermo Ortiz Jimenez

Years of a fierce competition have naturally selected the fittest deep learning algorithms. Yet, although these models work well in practice, we still lack a proper characterization of why they do so. This poses serious questions about the robustness, trus ...

EPFL2023

Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks

Rahul Parhi

We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2023

Tree-cotree decomposition of isogeometric mortared spaces in H(curl) on multi-patch domains

Rafael Vazquez Hernandez, Bernard Kapidani

When applying isogeometric analysis to engineering problems, one often deals with multi-patch spline spaces that have incompatible discretisations, e.g. in the case of moving objects. In such cases mortaring has been shown to be advantageous. This contribu ...

2022

Fourier uniqueness and interpolation in Euclidean space

Martin Peter Stoller

We prove that every Schwartz function in Euclidean space can be completely recovered given only its restrictions and the restrictions of its Fourier transform to all origin-centered spheres whose radii are square roots of integers. In particular, the only ...

EPFL2022

Perturbed Fourier uniqueness and interpolation results in higher dimensions

Martin Peter Stoller, Joao Pedro Gonçalves Ramos

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform ...

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

Fourier Interpolation From Spheres

Martin Peter Stoller

In every dimension d >= 2, we give an explicit formula that expresses the values of any Schwartz function on R-d only in terms of its restrictions, and the restrictions of its Fourier transform, to all origin-centered spheres whose radius is the square roo ...

2021

Perceptual effects of daylight patterns in architecture

Kynthia Chamilothori

The profound impact of daylight on occupants is broadly recognized in the fields of architecture and lighting. In particular, the façade and its interplay with light is a central design element, while the diversity of daylight in space is widely acknowledg ...

EPFL2019

Fourier interpolation on the real line

Maryna Viazovska

In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0,+/- 1,+/- ...

2019

Neural Tanget Kernel: Convergence and Generalization in Neural Networks

Clément Hongler, Franck Raymond Gabriel, Arthur Ulysse Jacot-Guillarmod

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit [12, 9], thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: ...

NEURAL INFORMATION PROCESSING SYSTEMS (NIPS)2018

Multiplicity Of Solutions For Linear Partial Differential Equations Using (Generalized) Energy Operators

Jean-Philippe Lucien Montillet

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [ 15]. To do so, the author ha ...

Int Center Scientific Research & Studies2017

Stochastic partial differential equations driven by Lévy white noises : generalized random processes, random field solutions and regularity

Thomas Marie Jean-Baptiste Humeau

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered r ...

EPFL2017

Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem

Alfio Quarteroni, Luca Dede', Claudia Maria Colciago

In this paper, we propose the mathematical and finite element analysis of a second-order partial differential equation endowed with a generalized Robin boundary condition which involves the Laplace--Beltrami operator by introducing a function space $H^1(\O ...

Society for Industrial and Applied Mathematics2015

On the stability of travelling waves with vorticity obtained by minimization

Boris Buffoni

We modify the approach of Burton and Toland Comm. Pure Appl. Math. LXIV. 975-1007 (2011) to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the functio ...

Springer Basel2013

Fourier Analysis Of Stationary Time Series In Function Space

Victor Panaretos, Shahin Tavakoli

We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density operator, which gen ...

Institute of Mathematical Statistics2013

On the Role of Constraints in Optimization under Uncertainty

Saurabh Anand Deshpande

This thesis addresses the problem of industrial real-time process optimization that suffers from the presence of uncertainty. Since a process model is typically used to compute the optimal operating conditions, both plant-model mismatch and process disturb ...

EPFL2012

Left-inverses of fractional Laplacian and sparse stochastic processes

Michaël Unser

The fractional Laplacian (-Delta)(gamma/2) commutes with the primary coordination transformations in the Euclidean space Rd: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < gamma < d, its inver ...

Springer2012

The non-linear stochastic wave equation in high dimensions : existence, Hölder-continuity and Itô-Taylor expansion

Daniel Conus

The main topic of this thesis is the study of the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time. We are interested in questions of existence and uniqueness of so ...

EPFL2008