[Paper Review] On the composition and decomposition of positive linear operators III: A non-trivial decomposition of the Bernstein operator

Heiner Gonska, Margareta Heilmann+2 more|arXiv (Cornell University)|Apr 12, 2012

Approximation Theory and Sequence Spaces5 references14 citations

TL;DR

This paper presents a non-trivial decomposition of the classical Bernstein operator $B_n$ as $B_n = \bar{\mathbb{B}}_n \circ F_n$, where $\bar{\mathbb{B}}_n$ is the genuine Beta operator and $F_n$ is a non-positive linear operator with interesting spectral and moment properties. The key contribution is an explicit representation of $F_n$ via inverse Beta operators and Jacobi polynomials, along with a Voronovskaya-type asymptotic formula and a connection to the Cooper-Waldron conjecture.

ABSTRACT

The central problem in this technical report is the question if the classical Bernstein operator can be decomposed into nontrivial building blocks where one of the factors is the genuine Beta operator introduced by Mühlbach and Lupaş. We collect several properties of the Beta operator such as injectivity, the eigenstructure and the images of the monomials under its inverse. Moreover, we give a decomposition of the form $B_n = \bar{\mathbb{B}}_n \circ F_n $ where $F_n$ is a nonpositive linear operator having quite interesting properties. We study the images of the monomials under $F_n$, its moments and various representations. Also an asymptotic formula of Voronovskaya type for polynomials is given and a connection with a conjecture of Cooper and Waldron is established. In an appendix numerous examples illustrate the approximation behaviour of $F_n$ in comparison to $B_n$.

Motivation & Objective

• To investigate whether the classical Bernstein operator $B_n$ can be decomposed into non-trivial positive linear operators.
• To explore the possibility of expressing $B_n$ as a composition $\bar{\mathbb{B}}_n \circ F_n$, where $\bar{\mathbb{B}}_n$ is the genuine Beta operator.
• To characterize the operator $F_n$ such that $B_n = \bar{\mathbb{B}}_n \circ F_n$, despite $F_n$ not being positive.
• To analyze the eigenstructure, moments, and inverse properties of the Beta operator $\bar{\mathbb{B}}_n$.
• To establish a connection between the asymptotic behavior of $F_n$ and the Cooper-Waldron conjecture on functionals related to approximation operators.

Proposed method

• The decomposition $B_n = \bar{\mathbb{B}}_n \circ F_n$ is constructed using the inverse of a differential operator $U_n$, defined via a sum involving $\widetilde{D}^{2l}$, which acts on monomials and eigenpolynomials.
• The operator $F_n$ is explicitly represented as $F_n f = \sum_{k=0}^n \nu_k^{(n)}(B_n f) p_k$, where $p_k$ are eigenpolynomials of $U_n$ and $\nu_k^{(n)}$ are linear functionals derived from inner products with Jacobi polynomials.
• The eigenpolynomials of $U_n$ are given by $p_k(x) = D^{k-2}[x^{k-1}(1-x)^{k-1}]$ for $k \geq 2$, with corresponding eigenvalues $\omega_k^{(n)} = \frac{(n-1)!n!}{(n-k)!(n+k-1)!}$.
• The inverse of $U_n$ is expressed as $U_n^{-1}p = \sum_{l=0}^{n-1} (-1)^l \frac{(n-1-l)!}{l!(n-1)!} \widetilde{D}^{2l}p$, enabling the construction of $F_n = U_n^{-1} B_n^2$.
• Moments of $F_n$ are analyzed through the relation $\int_0^1 b_{n-2,k-1}(t) \varphi_{n,i}(t) dt = \frac{1}{n-1} b_{n,i}(k/n)$, where $\varphi_{n,i}$ are the basis functions of $F_n$.
• Asymptotic behavior is studied by showing $\nu_k^{(n)}(f) \to \int_0^1 (f(t) - Lf(t)) J_{k-2}(t) dt$ as $n \to \infty$, with $Lf(x) = (1-x)f(0) + x f(1)$.

Experimental results

Research questions

• RQ1 Can the classical Bernstein operator $B_n$ be non-trivially decomposed into a composition of the genuine Beta operator $\bar{\mathbb{B}}_n$ and another operator $F_n$?
• RQ2 What are the spectral and moment properties of the non-positive operator $F_n$ that satisfies $B_n = \bar{\mathbb{B}}_n \circ F_n$?
• RQ3 How do the functionals $\nu_k^{(n)}$ associated with $F_n$ behave asymptotically as $n \to \infty$?
• RQ4 Is there a connection between the asymptotic behavior of $F_n$ and the Cooper-Waldron conjecture on approximation functionals?
• RQ5 What is the explicit representation of $F_n$ in terms of Jacobi polynomials and inverse differential operators?

Key findings

• The Bernstein operator admits a non-trivial decomposition as $B_n = \bar{\mathbb{B}}_n \circ F_n$, where $F_n$ is a non-positive linear operator, establishing a novel factorization of $B_n$.
• The inverse of the differential operator $U_n$ is explicitly given by $U_n^{-1}p = \sum_{l=0}^{n-1} (-1)^l \frac{(n-1-l)!}{l!(n-1)!} \widetilde{D}^{2l}p$, which allows the construction of $F_n = U_n^{-1} B_n^2$.
• The eigenpolynomials of $U_n$ are $p_k(x) = D^{k-2}[x^{k-1}(1-x)^{k-1}]$ for $k \geq 2$, with eigenvalues $\omega_k^{(n)} = \frac{(n-1)!n!}{(n-k)!(n+k-1)!}$, and these are shared by the operators $\widetilde{D}^{2l}$.
• The functionals $\nu_k^{(n)}$ are explicitly characterized as $\nu_0^{(n)}(f) = f(0)$, $\nu_1^{(n)}(f) = f(1) - f(0)$, and for $k \geq 2$, $\nu_k^{(n)}(f) = (\omega_k^{(n)})^{-1} \int_0^1 (B_n f(t) - Lf(t)) J_{k-2}(t) dt$.
• As $n \to \infty$, $\nu_k^{(n)}(f) \to \int_0^1 (f(t) - Lf(t)) J_{k-2}(t) dt$, confirming convergence to the functionals in the Cooper-Waldron conjecture.
• The asymptotic formula for $F_n f(x)$ is $F_n f(x) = Lf(x) + \sum_{k=2}^n x(1-x) J_{k-2}(x) \frac{(n-k)!(n+k-1)!}{(n-1)!n!} \int_0^1 (B_n^2 f(t) - Lf(t)) J_{k-2}(t) dt$, with pointwise convergence to the limit involving $f(t) - Lf(t)$.