A nonstandard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered, and their respective computational complexities are analyzed. Although the techniques employed are usually asociated to numerical analysis, in this case all the computations are carried out using exact rational arithmetic. The power of the Kronecker product of matrices in this application is stressed.
In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extens…
We provide a general framework to construct fractal interpolation surfaces (FISs) for a prescribed countably infinite data set on a rectangular grid. Using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operator are established, thereby initiating the interaction of the notion of fractal interpolation with the theory of nonlinear operators.
Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases
School of Mathemathics, Key Lab. of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University（吉林大学数学学院，符号计算与知识工程教育部重点实验室）
This paper investigates some univariate and bivariate constrained interpolation problems using rational quartic fractal interpolation functions, which has been submitted long back in a reputed journal and revised as per the journal requirement. This research is extension of the work [S. K. Katiyar and A. K. B. Chand, Shape Preserving Rational Quartic Fractal Functions, Fractal, in Press].
University of Basilicata（巴西利卡塔大学）；C.N.R. National Research Council of Italy（意大利国家研究委员会）
We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. The resized image is achieved by sampling a bivariate polynomial which globally interpolates the data at the new scale. The method's particularities lay in both the sampling model and the interpolation polynomial we use. Rather than classical uniform grids, we consider an unusual sampling system based on Chebyshev zeros of the first kind. Such optimal distribution of nodes permits to consider near--best interpolation polynomials defined by a filter of de la Vallée Poussin ty…
Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we prove that there are at most seven linearly independent curves of degree less than or equal to $k$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly seven such curves. Namely, we prove that then the set $\mathcal X$ has a very special construction: all its nodes but three belong to a (maximal) curve of degr…
