相关论文: An essay on some problems of approximation theory

Potential functional approximations are an intriguing alternative to density functional approximations. The potential functional that is dual to the Lieb density functional is defined and properties given. The relationship between…

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…

This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These…

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

The Langevin algorithms are frequently used to sample the posterior distributions in Bayesian inference. In many practical problems, however, the posterior distributions often consist of non-differentiable components, posing challenges for…

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^{2}$ exists, and $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2\pi i n \alpha} \Big | d \alpha\leq N^{o(1)}$$ as $N \rightarrow \infty$, then…

This article is a review of functional $f(R)$ approximations in the asymptotic safety approach to quantum gravity. It mostly focusses on a formulation that uses a non-adaptive cutoff, resulting in a second order differential equation. This…

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…

A longstanding open problem in the intersection of group theory and operator algebras is whether all groups are MF, that is, approximated by asymptotic representations with respect to the operator norm. More generally, for $1 \leq p \leq…

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…

For the class of de Branges-Rovnyak spaces $\mathcal{H}(b)$ of the unit disk $\mathbb{D}$ defined by extreme points $b$ of the unit ball of $H^\infty$, we study the problem of approximation of a general function in $\mathcal{H}(b)$ by a…

The paper concerns the uniform polynomial approximation of a function $f$, continuous on the unit Euclidean sphere of ${\mathbb R}^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we…

We find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar L\'evy-type stochastic processes. We derive rigorous error bounds for the…

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function,…

This paper concerns the universal approximation property with neural networks in variable Lebesgue spaces. We show that, whenever the exponent function of the space is bounded, every function can be approximated with shallow neural networks…

Modern applications require methods that are computationally feasible on large datasets but also preserve statistical efficiency. Frequently, these two concerns are seen as contradictory: approximation methods that enable computation are…

Although an input distribution may not majorize a target distribution, it may majorize a distribution which is close to the target. Here we introduce a notion of approximate majorization. For any distribution, and given a distance $\delta$,…

We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups:…

We show that the Laplace approximation of a supremum by Lp-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual appear naturally when using this…