A new class of positive linear operators preserving logarithmic functions

TL;DR

The paper develops a new class of positive linear operators $L_n$ that preserve the logarithmic transform $ln_mu(x)$ and relate to Bernstein/King-type frameworks. It proves pointwise and uniform convergence, provides a Voronovskaja-type asymptotic formula, and derives saturation and inverse theorems tied to a second-order differential operator, along with shape-preserving and BV-boundedness properties. A key payoff is the log-preservation property which enables a simple, constructive approach to mitigates multiplicative noise in signals, demonstrated via a toy signal-denoising application. The work broadens constructive approximation tools by integrating logarithmic reproduction with classical polynomial- and exponential-based operator theory, offering potential applications in log-linearization of nonlinear transformations and despeckling tasks.

Abstract

In this paper, we introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that reproduce the logarithmic function $\ln(1+μ+x)$, with $μ> 0$ and $x \in [0,1]$. We prove pointwise and uniform convergence and we derive a quantitative estimate of the approximation error in terms of the modulus of continuity. We also obtain a Voronovskaja-type asymptotic formula, that is used to establish saturation results and inverse theorems. In particular, the saturation class of the considered approximation process is characterized by solving a second order differential equation. Shape-preserving properties, such as monotonicity, concavity and variation diminishing, are also investigated. Finally, a simple application to signal denoising is addressed.

Figures (4)

• Figure 1: Graphs of the functions $a_n(x)$ for $n=2,4,6$ compared with the identity function $e_1(x)=x$. The functions $a_n(x)$ approximate $e_1$ from above.
• Figure 2: The (blue) solid line represents the original signal $f$, the dotted (red) plot represents the signal $g$ affected by the multiplicative noise, the asterisk (yellow) plot represent the denoised signal by (\ref{['denoising-method']}) for $\mu_1$, with $n=10$ (on the left) and $n=30$ (on the right). The maximum reconstruction errors are $0.1109$ and $0.0343$, respectively.
• Figure 3: The (blue) solid line represents the original signal $f$, the dotted (red) plot represents the signal $g$ affected by the multiplicative noise, the asterisk (yellow) plot represent the denoised signal by (\ref{['denoising-method']}) for $\mu_2$, with $n=10$ (on the left) and $n=30$ (on the right). The maximum reconstruction error are $0.0658$ and $0.0202$, respectively.
• Figure 4: The (blue) solid line represents the original signal $f$, the dotted (red) plot represents the signal $g$ affected by the multiplicative noise, the asterisk (yellow) plot represent the denoised signal by (\ref{['denoising-method']}) for $\mu_3$, with $n=10$ (on the left) and $n=30$ (on the right). The maximum reconstruction errors are $0.0622$ and $0.0191$, respectively.

Theorems & Definitions (22)

• Theorem 2.1: ConApp
• Lemma 3.1
• proof
• Theorem 4.1
• proof
• Definition 4.1
• Proposition 4.1: Altomare2010
• Proposition 4.2
• proof
• Theorem 5.1