Related papers: Optimal sampling patterns for Zernike polynomials
We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are…
We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of…
Image resampling is a necessary component of any operation that changes the size of an image or its geometry. Methods tuned for natural image upsampling (roughly speaking, image enlargement) are analyzed and developed with a focus on their…
We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Pad\'e approximation problem in [Van Barel and…
Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gr\"obner-basis or resultant-based methods due…
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted…
We introduce and investigate classes of normed or quasinormed distribution spaces of generalized smoothness that can be obtained by various interpolation methods applied to classical Sobolev, Nikolskii-Besov, and Triebel-Lizorkin spaces. An…
This paper investigates the spectral norm version of the column subset selection problem. Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a positive integer $k\leq\text{rank}(\mathbf{A})$, the objective is to select exactly $k$…
Neville's algorithm is known to provide an efficient and numerically stable solution for polynomial interpolations. In this paper, an extension of this algorithm is presented which includes the derivatives of the interpolating polynomial.
The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we…
Using factorisation and Arov-Krein inequality results, we derive important inequalities (in terms of $S$-nodes) in interpolation problems.
Topology optimization is one of the engineering tools for finding efficient design. For the material interpolation scheme, it is usual to employ the SIMP (Solid Isotropic Material with Penalization) or the homogenization based interpolation…
We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has…
Spatial mode sorting has come to prominence as an optical processing modality capable of saturating fundamental limits to numerous sensing tasks including wavefront sensing, coronagraphy, and superresolution imaging. But despite their…
We develop a local polynomial spline interpolation scheme for arbitrary spline order on bounded intervals. Our method's local formulation, effective boundary considerations and optimal interpolation error rate make it particularly useful…
Generative models based on normalizing flows are very successful in modeling complex data distributions using simpler ones. However, straightforward linear interpolations show unexpected side effects, as interpolation paths lie outside the…
We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
The influence of low-spatial frequency errors of an optical component of an imaging system on the point spread function can be quantified using Zernike polynomials. High-spatial frequency errors cause strong scattering due to which the…