טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentUri Itai
SubjectOn the Eigenstructure of the Bernstein Kernel
DepartmentDepartment of Mathematics
Supervisor Professor Emeritus Ziegler Zvi


Abstract

The Bernstein operator and Bernstein polynomial basis are both intrinsic parts of approximation theory. While various aspects of the Bernstein kernel have been studied using different approaches, only a few papers have employed linear algebraic methods to investigate the kernel. In this paper, we describe the Bernstein kernel's linear algebraic properties and their relationships to the general properties of the Bernstein kernel.
The Bernstein operator matrix and the transformation matrix from the standard monomial basis to the Bernstein basis are both totally positive and lower triangular but have very different eigenstuctures.

The non-diagonalizability of the transformation matrix from the standard basis to the Bernstein basis is a feature that exists in other cases as well. In fact, we will show that two other transformation matrices, the Wilkinson polynomial basis and the reduced Bernstein polynomial basis exhibit similar behavior.

The Bernstein operator has many variants, two of which we will describe here. The first is the Bernstein-Kantorovich operator and  the Bernstein-Durrmeyer operator, which are both  approximation operators .

 We will use the SVD matrix technique for both the transformation matrix and the operator matrix, where both of the two matrices are positive definite.

The connection between a transformation matrix of a given polynomial basis and the linear operator which is induced is explored.
Two  additional polynomial transformations matrices would be explore the transformation matrices from the Legendre basis to both the Bernstein basis and the Stancu Basis. The transformation matrix from the Legendre basis to the Bernstein basis has some interesting properties .