Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.05434

Bivariate Bernstein Fractal Interpolation and Numerical Integration on Triangular Domains

Aparna M. P.·P. Paramanathan
View original source ↗Read source document ↗Export metadata ↓

Abstract

The fundamental aim of this paper is to provide the approximation and numerical integration of a discrete set of data points with Bernstein fractal approach. Using Bernstein polynomials in the iterated function system, the paper initially proposes the numerical integration formula for the data set corresponding to univariate functions. The proposed formula of integration is shown to be convergent by examining the data sets of certain weierstrass functions. The paper then extends the Bernstein fractal approximation and numerical integration technique to two dimensional interpolating regions. Bernstein polynomials defined over triangular domain has been used for the purpose. The triangular domain has been partitioned and the newly generated points are assigned colors in a particular manner to maintain the chromatic number as 3. Following the above mentioned construction and approximation of bivariate Bernstein fractal interpolation functions, the paper introduces the numerical double integration formula using the constructed functions. The convergence of the double integration formula towards the actual integral value of the data sets is displayed with the help of some examples including the benchmark functions. Both the newly introduced iterated function systems are verified for their hyperbolicity and the resultant fractal interpolation functions are shown to be continuous.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Aparna M. P., P. Paramanathan. 2024-02-08. Bivariate Bernstein Fractal Interpolation and Numerical Integration on Triangular Domains. https://arxiv.org/abs/2402.05434

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗