Related papers: Multivariate Lagrange Interpolation and an Applica…
We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete…
In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley-Bacharach condition. In particular, by bounding the number of points satisfying the Cayley-Bacharach condition, we force them to…
High-dimensional Lagrange interpolation plays a pivotal role in finite element methods, where ensuring the unisolvence and symmetry of its interpolation space and nodes set is crucial. In this paper, we leverage group action and group…
In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that…
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…
Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node…
This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate…
In this paper, we study the geometry of points in complex projective space that satisfy the Cayley-Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez…
We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the non-compactness of the boundary,…
In this article, we use the inverse function theorem for Banach spaces to interpolate a given real analytic spacelike curve $a$ in Lorentz-Minkowski space $\mathbb{L}^3$ to another real analytic spacelike curve $c$, which is ``close" enough…
In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank $1$ and are generalizations of the well-known Padua points. We show that…
This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and…
The maximum volume principle is investigated as a means to solve the following problem: Given a set of arbitrary interpolation nodes, how to choose a set of polynomial basis functions for which the Lagrange interpolation problem is…
It is known that every function with a finite support over a given field can be interpolated by means of the Lagrangian polynomial. The question is if a similar interpolation is possible if one considers a unitary ring or a Boolean algebra…
Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of…
Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution…
Given subvarieties $X, Y$ of a complex algebraic variety $S$ of complementary dimension, must they intersect? When $S$ is projective space, this is a consequence of the classical B\'ezout theorem, and an analogue for simple abelian…
The classical polynomial interpolation problem in several variables can be generalized to the case of points with greater multiplicities. What is known, as yet, is essentially concentrated in the Alexander-Hirschowitz Theorem which says…