Investigation of conic equations under bernstein operators

One of the most important problems in approximation theory in mathematical analysis is the determination of sequences of polynomials that converge to functions and have the same geometric properties. This type of approach is called the shape-preserving approximation. These types of problems are usually handled depending on the convexity of the functions, the degree of smoothness depending on the order of differentiability, or whether it satisfies a functional equation. The problem addressed in this thesis belongs to the third class. A quadratic bivariate algebraic equation denotes geometrically some well-known shapes such as circle, ellipse, hyperbola and parabola. Such equations are known as conic equations. In this study, it is investigated whether conic equations transform into a conic equation under bivariate Bernstein polynomials, and if so, which conic equation it transforms into. It was observed that the single-index bivariate Bernstein polynomials preserved the shapes determined by the conic equations, while the double-index ones preserved in some special cases.