M.Sc Thesis | |

M.Sc Student | Itai Uri |
---|---|

Subject | On the Eigenstructure of the Bernstein Kernel |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Zvi Ziegler |

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 .