Related papers: A Stone-Weierstrass approximation theorem for mono…
The s-th forward difference sequence that tends to zero, inspired by the consecutive terms of a sequence approaching zero, is examined in this study. Functions that take sequences satisfying this condition to sequences satisfying the same…
We provide a general theorem on the asymptotic behavior of stochastic processes that conform to a relaxed supermartingale condition. The distinguishing feature of our result is that it provides quantitative convergence guarantees at a much…
We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as…
Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindel\"of locally compact Hausdorff space are given, and used to prove Stone-Weierstra{\ss}-type theorems…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application…
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 show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in the whole space or in any cylindrical smooth domain with smooth boundary data one can find an…
We consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best…
We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the…
We develop the theory of invariant structure preserving and free functions on a general structured topological space. We show that an invariant structure preserving function is pointwise approximiable by the appropriate analog of…
We study feedforward neural networks with inputs from a topological space (TFNNs). We prove a universal approximation theorem for shallow TFNNs, which demonstrates their capacity to approximate any continuous function defined on this…
Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly…
We generalize a version of Lavrent\'ev's theorem which says that a function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial…
The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…
We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain $[0,1]$ is physically desired, these polynomial families…
In this short note we give a proof of the refined version of the uniform invariant approximation property for compact (non-commutative) groups following the Bourgain's approach.
We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness…