Properties of Newton polynomials and Toeplitz operators on Newton spaces

Abstract

In this paper, we study properties of Toeplitz operators on the Newton space $N^2({\mathbb H})$ which has Newton polynomials as an orthonormal basis. We show that for $\textbf{N}=(N_0,N_1,\ldots , N_n)^T$ and $\textbf{m}=(1,z,\ldots , z^n)^T$, the equation

$$\begin{aligned} \textbf{V}\textbf{U}\textbf{N}=\textbf{m} \end{aligned}$$

is the transformations between the basis functions which map monomials to Newton polynomials where $\textbf{V}$ and $\textbf{U}$ are given as in Theorem 2.1. Moreover, we consider the truncated Toeplitz operator on $N^{2}({\mathbb H}).$