Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.Comment: 29 page

A BEM based on the Bézier/Bernstein polynomial for acoustic waveguide modelization

42nd International Conference on Boundary Elements and other Mesh Reduction Methods, BEM/MRM 2019; ITeCons-University of CoimbraCoimbra; Portugal; 2 July 2019 through 4 July 2019; Code 155806. Publicado en WIT Transactions on Engineering Sciences, Vol 126This paper proposes a novel boundary element approach formulated on the Bézier–Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aided design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton–Bernstein algorithm. The applicability of the proposed method is demonstrated by solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.Ministerio de Economía y Competitividad BIA2016-75042-C2-1-

Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

We prove that both Polynomial Calculus and Sums-of-Squares proof systems admit a strong form of feasible interpolation property for sets of polynomial equality constraints. Precisely, given two sets P(x,z) and Q(y,z) of equality constraints, a refutation ? of P(x,z) ? Q(y,z), and any assignment a to the variables z, one can find a refutation of P(x,a) or a refutation of Q(y,a) in time polynomial in the length of the bit-string encoding the refutation ?. For Sums-of-Squares we rely on the use of Boolean axioms, but for Polynomial Calculus we do not assume their presence

Про інтерполяцію функції двох змінних в обмеженій області за її значеннями на множині кривих, заданих параметрично

Розглянуто задачу iнтерполяцiї функцiї двох змiнних в обмеженiй областi за вiдомими значеннями на множинi кривих, якi заданi параметрично. За допомогою теорiї операторного iнтерполювання побудовано операторний полiном, який має вiдповiднi iнтерполяцiйнi властивостi. Наведено приклади чисельних експериментiв.Рассмотрена задача интерполяции функции двух переменных в ограниченной области по известным значениям на множестве кривых, которые заданы параметрически. С помощью теории операторного интерполирования построен операторный полином, владеющий соответствующими интерполяционными свойствами. Приведены примеры численных экспериментов.We consider the interpolation problem for a function of two variables in a bounded domain from the given values on a set of curves with parametric representation. On the basis of the theory of operator interpolation, the operator polynomial, which has corresponding interpolation properties, is constructed. The examples of numerical experiments are presented

Orbit Determination with the two-body Integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.Comment: 23 pages, 1 figur

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domainMinisterio de Economía y Competitividad BIA2016-75042-C2-1-RFondos FEDER POCI-01-0247-FEDER-01775

Kajian Interpolasi Dua Dimensi dalam Tabel Nilai Kritik Sebaran F Berbantuan Program Matlab

The purposes of this research are to examine how to perform two-dimensional interpolation for determine the value of the F distribution, to make a two-dimensional interpolation program using Matlab and reviewing the comparison of the methods used (manually and program). This research was conducted by using literature study approach. The results of this research are:  first, the two-dimensional interpolation in F distribution table can be done using the successive univariate polynomial interpolation. Two-dimensional interpolation formulas can be made by referring to the general form of Lagrange and Newton's interpolation polynomials. Second, a two-dimensional interpolation program assisted by Matlab that is a program that can determine the intermediate value of a function in two variables using the Lagrange and Newton’s polynomial interpolation formula has been created. Third, based on the final results, there is no difference shown by the two methods used. Judging from the process, two-dimensional interpolation using the Lagrange polynomial method has advantages in simplicity of programming, but requires a long time in manual completion. While the Newton polynomial interpolation method has advantages in the simplicity of the manual work process, but it requires a long time to make the program.Tujuan penelitian ini adalah mengkaji cara melakukan interpolasi dua dimensi untuk menentukan nilai tabel sebaran F, membuat  program interpolasi dua dimensi berbantuan Matlab, dan mengkaji perbandingan dari metode-metode yang digunakan (manual maupun berbantuan program). Jenis penelitian ini adalah penelitian dasar dengan pendekatan studi literatur. Karena itu keseluruhan data penelitian diambil dari buku-buku dan referensi lain yang relevan dengan masalah yang dikaji. Adapun hasil penelitian adalah sebagai berikut: pertama, interpolasi dua dimensi dalam tabel nilai kritik sebaran F dapat dilakukan dengan menggunakan prosedur interpolasi satu variabel secara berurutan. Formula interpolasi dua dimensi dapat dibuat dengan mengacu pada bentuk umum polinom interpolasi Lagrange dan Newton. Kedua, telah dibuat suatu program interpolasi dua dimensi berbantuan Matlab yaitu program yang dapat menentukan nilai tengahan suatu fungsi dalam dua variabel dengan menggunakan formula interpolasi polynomial Lagrange dan formula interpolasi polynomial Newton. Ketiga, dilihat dari hasil akhir, tidak ada perbedaan yang ditunjukkan oleh kedua metode yang digunakan. Dilihat dari proses, interpolasi dua dimensi dengan menggunakan metode interpolasi polynomial Lagrange memiliki kelebihan dalam kesederhanaan pembuatan program, namun memerlukan waktu yang cukup lama dalam penyelesaian manualnya. Sedangkan metode interpolasi polynomial Newton memiliki kelebihan dalam kesederhanaan proses kerja secara manual, namun memerlukan waktu yang cukup lama dalam pembuatan progra