Related papers: Minimal Cubature rules and polynomial interpolatio…
We consider a sequence of composite bivariate Bernstein operators and the cubature formula associated with them. The upper bounds for the remainder term of the cubature formula are described in terms of moduli of continuity of order two.…
In this work we develop the Gaussian quadrature rule for weight functions involving fractional powers, exponentials and Bessel functions of the first kind. Besides the computation based on the use of the standard and the modified Chebyshev…
We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(\lambda)}(x))^2(1-x)^\alpha (1+x)^\beta\, dx, \end{equation*} where $C_n^{(\lambda)}$ denotes the Gegenbauer-polynomial of index $\lambda>0$ and $\alpha,\beta>-1$. We give…
The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for finite discrete…
We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovi\v{c}s, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature…
The $\mathcal{L}_2$ discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Wo\'{z}niakowski to…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
We investigate quadrature rules for measures supported on real algebraic and rational curves, focusing on the {odd-degree} case \(2s-1\). Adopting an optimization viewpoint, we minimize suitable penalty functions over the space of…
The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on…
In this paper we obtain $L^1$-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In…
In this paper we are dealing with two classes of mean curvature type problems that generalize the translating soliton problem. A first result proves that the solutions to these problems have unique interior critical points. Using this…
To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the…
In this contribution we introduce a mixed interpolation-regression operator for functions defined in some domains of the plane. We focus the attention on the ellipse, an annulus and a polygon. An upper bound for such an operator is…
We completely determine the minimal polynomial of an arbitrary simple highest weight module $L(\lambda)$ over a complex classical Lie algebra $\mathfrak{g}\subseteq\mathfrak{gl}_N$ relative to its defining module $\pi=\mathbb{C}^{N}$. These…
A Chebyshev-type quadrature for a given weight function is a quadrature formula with equal weights. In this work we show that a method presented by Kane may be used to produce tight bounds for the minimal number of nodes required in…
A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by…
For the weight function $W_\mu(x) = (1-|x|^2)^\mu$, $\mu > -1$, $\lambda > 0$ and $b_\mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ \la f,g \ra = {b_\mu…
Positive interpolatory cubature formulas (CFs) are constructed for quite general integration domains and weight functions. These CFs are exact for general vector spaces of continuous real-valued functions that contain constants. At the same…
Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by…
We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special $R_{II}$ recurrence relation. We also look into some methods for generating the nodes (which lie on…