We introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that preserve the logarithmic function ln(1 + \mu + x), with \mu > 0 and x is an element of [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 Voronovskajatype 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.
A NEW CLASS OF POSITIVE LINEAR OPERATORS PRESERVING LOGARITHMIC FUNCTIONS
Angeloni, L
;Costarelli, D;Darielli, C
2026
Abstract
We introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that preserve the logarithmic function ln(1 + \mu + x), with \mu > 0 and x is an element of [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 Voronovskajatype 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
