Related papers: $C^{\infty}$ rational approximation and quasi-hist…
Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently.
We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the…
This paper presents a new approach of constructing $\alpha$-fractal interpolation functions (FIFs) using neural network operators, integrating concepts from approximation theory. Initially, we construct $\alpha$-fractals utilizing neural…
Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals…
The mapped bases or Fake Nodes Approach (FNA), introduced in [10], allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied in preventing the appearance of the Gibbs phenomenon…
We propose a physics-based regularization technique for function learning, inspired by statistical mechanics. By drawing an analogy between optimizing the parameters of an interpolator and minimizing the energy of a system, we introduce…
Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting,…
A method is suggested for interpolating between small-variable and large-variable asymptotic expansions. The method is based on self-similar approximation theory resulting in self-similar root approximants. The latter are more general than…
A \Riesz-basis sequence for $L_2[-\pi,\pi]$ is a strictly increasing sequence $X:=(x_j)_{j\in\mathbb{Z}}$ in $\mathbb{R}$ such that the set of functions $\left(e^{-ix_j(\cdot)}\right)_{j\in\mathbb{Z}}$ is a Riesz basis for $L_2[-\pi,\pi]$.…
The calculation and manipulation of large multi-variable rational functions is a key bottleneck in multi-loop calculations. In these conference proceedings, based on my article [Chawdhry (2023) arXiv:2312.03672], I present a technique to…
The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…
We design a quasi-interpolation operator from the Sobolev space $H^1_0(\Omega)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1)…
We consider the following interpolation problem. Suppose one is given a finite set $E \subset \mathbb{R}^d$, a function $f: E \rightarrow \mathbb{R}$, and possibly the gradients of $f$ at the points of $E$. We want to interpolate the given…
[104] introduced a powerful \emph{fully lifted} (fl) statistical interpolating mechanism. It established a nested connection between blirps (bilinearly indexed random processes) and their decoupled (linearly indexed) comparative…
Soft extrapolation refers to the problem of recovering a function from its samples, multiplied by a fast-decaying window and perturbed by an additive noise, over an interval which is potentially larger than the essential support of the…
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^\alpha$ for $\alpha \in…
Fractal interpolation functions (FIFs) developed through iterated function systems (IFSs) prove more versatile than classical interpolants. However, the applications of FIFs in the domain of `shape preserving interpolation' are not fully…
Craig interpolation is a fundamental property of classical and non-classic logics with a plethora of applications from philosophical logic to computer-aided verification. The question of which interpolants can be obtained from an…
We introduce an interpolation--regression operator for polynomial approximation on the unit sphere $\mathbb{S}^2$ from discrete samples. The approximant is a spherical polynomial of degree $r$ which interpolates the data on a prescribed…
A powerful statistical interpolating concept, which we call \emph{fully lifted} (fl), is introduced and presented while establishing a connection between bilinearly indexed random processes and their corresponding fully decoupled (linearly…