Generalized Stechkin-Marchaud Inequalities via Deep Neural Network Approximation: Theorems on Convergence, Stability, and Error Bounds

Nagham Ali  Hussen

Authors

Nagham Ali Hussen University of Information technology and communications, College of Engineering, Mobile Computing and Communications engineering, Baghdad, Iraq

Keywords:

Stechkin-Marchaud inequality, Deep neural networks, Approximation theory, Modulus of smoothness, K-functional

Abstract

In this paper, an extension of the classical and well-known Stechkin-Marchaud inequality model based on presenting standardized theorems functionalities for deep neural network (DNN) based approximations in the -spaces where . Modeling on the operator approaches and modulus of the property of smoothness. In this paper, three powerful results are illustrated: (i) a convergence theorem illustrating the uniform based approximation power of the neural operators, (ii) a stability theorem consisting the robustness property over perturbations, and (iii) an obvious error bound-theorem for the linking network-complexity in order to improve the approximation quality. The proposed approach distinct the equality between two modules named K-functionals and smoothness, producing an accurate foundation for neural network approximators in the practical spaces. These findings of the theorem help for practical applications in the fields of data-driven scientific and signal-processing. Furthermore, validating of the numerical of the theoretical bounds and focusing the performance enhancements via classical operators. This paper sets a basic for bridging approximation and recent deep learning models.

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