Related papers: Shearlet approximation of functions with discontin…
Taylor's formula holds significant importance in function representation, such as solving differential difference equations, ordinary differential equations, partial differential equations, and further promotes applications in visual…
We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
Discontinuities in spatial derivatives appear in a wide range of physical systems, from creased thin sheets to materials with sharp stiffness transitions. Accurately modeling these features is essential for simulation but remains…
We use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the…
By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…
Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was…
We discuss the applicability of quasilinear-type approximations for a turbulent system with a large range of spatial and temporal scales. We consider a paradigm fluid system of rotating convection with a vertical and horizontal temperature…
We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Sham density functional theory to open systems with fluctuating particle numbers. When used in the self-consistent Kohn-Sham procedure on…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a…
We investigate smooth approximations of functions, with prescribed gradient behavior on a distinguished stratified subset of the domain. As an application, we outline how our results yield important consequences for a recently introduced…
In this paper we study the strong convergence for the Euler-Maruyama approximation of a class of stochastic differential equations whose both drift and diffusion coefficients are possibly discontinuous.
Hierarchical neural networks are exponentially more efficient than their corresponding "shallow" counterpart with the same expressive power, but involve huge number of parameters and require tedious amounts of training. By approximating the…
Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…
Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of…
We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the…
A conjecture appears in \cite{milsteinscheme}, in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear…
Functional data analysis is ubiquitous in most areas of sciences and engineering. Several paradigms are proposed to deal with the dimensionality problem which is inherent to this type of data. Sparseness, penalization, thresholding, among…
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of H{\"o}lder continuous coefficients as well as piecewise smooth drifts with…