Related papers: Constrained mock-Chebyshev least squares approxima…
The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there…
The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge…
In computational practice, we often encounter situations where only measurements at equally spaced points are available. Using standard polynomial interpolation in such cases can lead to highly inaccurate results due to numerical…
Higher-degree polynomial interpolations carried out on uniformly distributed nodes are often plagued by {\it overfitting}, known as Runge's phenomenon. This work investigates Runge's phenomenon and its suppression in various versions of the…
Approximation theorem is one of the most important aspects of numerical analysis that has evolved over the years with many different approaches. Some of the most popular approximation methods include the Lebesgue approximation theorem, the…
The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee Poussin filters. These polynomials can be an useful device for many theoretical and…
Previous works show convergence of rational Chebyshev approximants to the Pad\'e approximant as the underlying domain of approximation shrinks to the origin. In the present work, the asymptotic error and interpolation properties of rational…
The interpolation-regression approximation is a powerful tool in numerical analysis for reconstructing functions defined on square or triangular domains from their evaluations at a regular set of nodes. The importance of this technique lies…
In this study, we conduct a thorough and meticulous examination of the Runge phenomenon. Initially, we engage in an extensive review of relevant literature, which aids in delineating the genesis and essence of the Runge phenomenon, along…
Chebyshev interpolation polynomials exhibit the exponential approximation property to analytic functions on a cube. Based on the Chebyshev interpolation polynomial approximation, we propose iterative polynomial approximation algorithms to…
For trigonometric series and series of Chebyshev polynomials, we defined trigonometric Hermite-Pad\'e and Hermite-Jacobi approximations, linear and nonlinear Hermite-Chebyshev approximations. We established criterion of the existence and…
The present paper concerns filtered de la Vall\'ee Poussin (VP) interpolation at the Chebyshev nodes of the four kinds. This approximation model is interesting for applications because it combines the advantages of the classical Lagrange…
Histopolation, or interpolation on segments, is a mathematical technique used to approximate a function $f$ over a given interval $I=[a,b]$ by exploiting integral information over a set of subintervals of $I$. Unlike classical polynomial…
We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted $\ell^1$ minimization has recently…
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 determine the pointwise error in Hermite interpolation by numerically solving an appropriate differential equation, derived from the error term itself. We use this knowledge to approximate the error term by means of a polynomial, which…
In this paper, we present a rigorous analysis of root-exponential convergence of Hermite approximations, including projection and interpolation methods, for functions that are analytic in an infinite strip containing the real axis and…
Chebyshev interpolation is a highly effective, intensively studied method and enjoys excellent numerical properties. The interpolation nodes are known beforehand, implementation is straightforward and the method is numerically stable. For…
We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of…
For a function that is analytic on and around an interval, Chebyshev polynomial interpolation provides spectral convergence. However, if the function has a singularity close to the interval, the rate of convergence is near one. In these…