[Paper Review] Geometric series of positive linear operators and inverse Voronovskaya theorem

Ulrich Abel, Mircea Ivan+1 more|arXiv (Cornell University)|Apr 21, 2013

Approximation Theory and Sequence Spaces17 references3 citations

TL;DR

This paper introduces geometric series of positive linear operators within a new class Λ, establishing convergence and deriving an inverse Voronovskaya theorem. It proves that for operators in Λ, the geometric series converges in the $ C_\psi[0,1] $ norm, and applies the results to Bernstein, Bernstein-Durrmeyer-type, and symmetrical Meyer-König and Zeller operators, yielding precise asymptotic expansions and error estimates.

ABSTRACT

We define the associated geometric series for a large class of positive linear operators and study the convergence of the series in the case of sequences of admissible operators. We obtain an inverse Voronovskaya theorem and we apply our results to the Bernstein operators, the Bernstein-Durrmeyer-type operators, and the symmetrical version of Meyer-König and Zeller operators.

Motivation & Objective

• To define and study geometric series of positive linear operators in a broad class of approximation operators preserving linear functions.
• To establish convergence of the geometric series $ G_L = \sum_{k=0}^\infty L^k $ for operators in class $ \Lambda $, where $ \|L(\psi)\|_\psi < 1 $.
• To prove an inverse Voronovskaya theorem characterizing the asymptotic behavior of iterates of such operators.
• To apply the general results to specific operators: Bernstein, Bernstein-Durrmeyer-type, and symmetrical Meyer-König and Zeller operators.
• To derive precise asymptotic expansions and error estimates for the operators' iterates and geometric series.

Proposed method

• Define the class $ \Lambda $ of positive linear operators $ L $ on $ C[0,1] $ that preserve linear functions, satisfy $ \|L(\psi)\|_\psi < 1 $, and are not the first-order Bernstein operator $ B_1 $.
• Introduce the space $ C_\psi[0,1] $ with norm $ \|f\|_\psi = \sup_{x\in(0,1)} |f(x)/\psi(x)| $, where $ \psi(x) = x(1-x) $, to control behavior near endpoints.
• Prove that $ L \in \Lambda $ implies $ L(C_\psi[0,1]) \subset C_\psi[0,1] $ and $ \|L\|_{\mathcal{L}(C_\psi)} = \|L(\psi)\|_\psi $.
• Establish convergence of the geometric series $ G_L = \sum_{k=0}^\infty L^k $ in the $ \|\cdot\|_\psi $ norm for $ L \in \Lambda $.
• Derive an inverse Voronovskaya theorem by analyzing the asymptotic expansion of $ Z_n(e_r)(x) $, showing $ Z_n(e_r)(x) = x^r + \frac{1}{n}\binom{r}{2}x^{r-1}(1-x)^2 + \varepsilon_{n,r}(x) $ with $ |\varepsilon_{n,r}(x)| \leq \frac{x(1-x)}{n(n-1)} C_r $.
• Use combinatorial identities and generating functions to analyze the moments of the operators, particularly $ m_n^r(x) $, and derive bounds involving $ \alpha_n(x) $ and error terms.

Experimental results

Research questions

• RQ1 Under what conditions does the geometric series $ \sum_{k=0}^\infty L^k $ converge for a positive linear operator $ L $?
• RQ2 Can an inverse Voronovskaya theorem be established for geometric series of positive linear operators in a general class?
• RQ3 How do the iterates of Bernstein, Bernstein-Durrmeyer-type, and symmetrical Meyer-König and Zeller operators behave asymptotically?
• RQ4 What is the precise asymptotic expansion of $ Z_n(e_r)(x) $, and what are the error bounds?
• RQ5 Can the convergence of the geometric series be controlled in the $ C_\psi[0,1] $ norm with explicit estimates?

Key findings

• The geometric series $ G_L = \sum_{k=0}^\infty L^k $ converges in the $ \|\cdot\|_\psi $ norm for all $ L \in \Lambda $, where $ \|L(\psi)\|_\psi < 1 $.
• For the operator $ Z_n $, the $ r $-th moment satisfies $ Z_n(e_r)(x) = x^r + \frac{1}{n}\binom{r}{2}x^{r-1}(1-x)^2 + \varepsilon_{n,r}(x) $ with $ |\varepsilon_{n,r}(x)| \leq \frac{x(1-x)}{n(n-1)} C_r $, where $ C_r $ depends only on $ r $.
• For $ r=2 $, the second moment satisfies $ m_n^2(x) = x(1-x)^2\left[\frac{1}{n} - \frac{1-2x}{n(n-1)} + \rho_n(x)\right] $ with $ |\rho_n(x)| \leq \frac{12}{n(n^2-1)} $.
• The operator $ Z_n^\star $ satisfies $ Z_n^\star(\psi|\alpha_n - \alpha_n(x)e_0|)(x) \leq \frac{\psi(x)}{n(n-1)}\left[\sqrt{\|\alpha_n\|} + \frac{12}{n+1}\right] $, confirming convergence in the required norm.
• The conditions of Theorems 4 and 5 are satisfied for $ (Z_n^\star)_{n \geq 3} $, ensuring the validity of the inverse Voronovskaya result.
• The results are applied to Bernstein, Bernstein-Durrmeyer-type, and symmetrical Meyer-König and Zeller operators, yielding explicit asymptotic expansions and error estimates.