Stancu-Type Generalizations of Neural Network Operators with Perturbed Sampling Nodes

Abstract

In this paper, we introduce a Stancu-type generalization of multivariate neural network operators by incorporating two parameters that perturb the sampling nodes. The proposed operators extend the existing neural network operator by allowing greater flexibility in the placement of sampling nodes. We establish the well-definedness and boundedness of the operators and prove uniform convergence on compact domains. Furthermore, quantitative error estimates are derived in terms of the modulus of continuity, leading to convergence rate results. Numerical experiments are presented to illustrate the approximation behavior of the proposed operators and to demonstrate the effect of the Stancu parameters on the sampling nodes and the approximation accuracy. Finally, the application of signal denoising is demonstrated using a synthetic ECG signal, showing that the proposed operators effectively suppress noise while preserving the signal's main characteristics.

Figures (4)

• Figure 1: Approximation of $f(s)$ using the Stancu-type NNOs for different values of $(\alpha,\beta)$ with $n=50$.
• Figure 2: Decay of the maximum approximation error for the Stancu-type NNOs as $n$ increases.
• Figure 3: Comparison between original sampling nodes $k/n$ and Stancu nodes $(k+\alpha)/(n+\beta)$.
• Figure 4: ECG signal denoising using the Stancu-type NNOs. The black curve represents the true ECG signal, the red curve corresponds to the noisy signal, and the blue curve shows the reconstructed (denoised) signal obtained by the proposed operator.

Theorems & Definitions (13)

• Definition 2.1
• Lemma 2.2
• Definition 3.1
• Remark 3.2
• Lemma 3.3: Well-definedness and boundedness
• proof
• Theorem 3.4: Uniform convergence
• proof
• Theorem 3.5: Rate of convergence
• proof