Some New Results Concerning the Classical Bernstein Cubature Formula

• Main Author: Dan Miclăuş
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Symmetry
• Subjects: Bernstein operator; quadrature formula; cubature formula; upper bound estimation
• Online Access: https://www.mdpi.com/2073-8994/13/6/1068

In this article, we present a solution to the approximation problem of the volume obtained by the integration of a bivariate function on any finite interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo><mo>×</mo><mo>[</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow></semantics></math></inline-formula>, as well as on any symmetrical finite interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo>−</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>]</mo><mo>×</mo><mo>[</mo><mo>−</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>]</mo></mrow></semantics></math></inline-formula> when a double integral cannot be computed exactly. The approximation of various double integrals is done by cubature formulas. We propose a cubature formula constructed on the base of the classical bivariate Bernstein operator. As a valuable tool to approximate any volume resulted by integration of a bivariate function, we use the classical Bernstein cubature formula. Numerical examples are given to increase the validity of the theoretical aspects.