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Simultaneous approximation by neural network operators with applications to Voronovskaja formulas

Marco Cantarini, Danilo Costarelli

TL;DR

This work addresses the simultaneous approximation of a function $f$ and its derivatives using neural network operators activated by sigmoidal functions. It develops a constructive framework, establishing uniform convergence of derivatives up to order $m$ on a subinterval $I_\delta$ and providing quantitative estimates via the modulus of continuity, together with Voronovskaja-type formulas that capture the exact high-order convergence through density-moment constants. The analysis hinges on truncated algebraic moments and Strang–Fix type conditions for the sigmoidal density $\,\phi_{\sigma}$, yielding explicit moment identities and enabling high-order results. Practical implications include precise derivative reconstruction from sampled data and potential applications in signal and image processing, with concrete examples (e.g., logistic activation) illustrating the theory.

Abstract

In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem for the derivatives of NN operators, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of the approximated derivative. Furthermore, a qualitative and quantitative Voronovskaja-type formula is established, which provides information about the high order of approximation that can be achieved by NN operators. To prove the above theorems, several auxiliary results involving sigmoidal functions have been established. At the end of the paper, noteworthy examples have been discussed in detail.

Simultaneous approximation by neural network operators with applications to Voronovskaja formulas

TL;DR

This work addresses the simultaneous approximation of a function and its derivatives using neural network operators activated by sigmoidal functions. It develops a constructive framework, establishing uniform convergence of derivatives up to order on a subinterval and providing quantitative estimates via the modulus of continuity, together with Voronovskaja-type formulas that capture the exact high-order convergence through density-moment constants. The analysis hinges on truncated algebraic moments and Strang–Fix type conditions for the sigmoidal density , yielding explicit moment identities and enabling high-order results. Practical implications include precise derivative reconstruction from sampled data and potential applications in signal and image processing, with concrete examples (e.g., logistic activation) illustrating the theory.

Abstract

In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem for the derivatives of NN operators, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of the approximated derivative. Furthermore, a qualitative and quantitative Voronovskaja-type formula is established, which provides information about the high order of approximation that can be achieved by NN operators. To prove the above theorems, several auxiliary results involving sigmoidal functions have been established. At the end of the paper, noteworthy examples have been discussed in detail.
Paper Structure (6 sections, 7 theorems, 90 equations, 1 figure)

This paper contains 6 sections, 7 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main definitions
  3. Auxiliary results
  4. Simultaneous approximation with applications to Voronovskaja formulas
  5. Particular cases and examples
  6. Final remarks, conclusions and future developments

Key Result

Theorem 2.1

Let $f: I \to \mathbb{R}$ be bounded. Then, at each point $x \in I$ where $f$ is continuous. Moreover, if $f$ is continuous on the whole $I$, we have:

Figures (1)

  • Figure 1: The Fourier transform of $\phi_{\sigma_{\ell}}$ (on the left) and its first derivative (on the right).

Theorems & Definitions (15)

  • Theorem 2.1: COSP1
  • Lemma 3.1: COVI2
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 4.1
  • proof
  • ...and 5 more