What question did this study set out to answer?

This research aims to establish effective multivariate approximations of continuous functions using neural networks in Banach spaces.

June 10, 2026Open Access

General multi-composite sigmoid based Banach space valued multivariate neural network approximations

GAGeorge A. AnastassiouUniversity of Memphis

Key Points

This research aims to establish effective multivariate approximations of continuous functions using neural networks in Banach spaces.
Developed multi-composite multivariate normalized neural network operators for function approximation.
Used Jackson type inequalities based on the modulus of continuity and Fréchet derivatives for analytic derivation.
Implemented neural networks with a single hidden layer employing multi-composite sigmoid activation functions.
Pointwise and uniform convergence of approximations is achieved.
Demonstrated efficiency in approximating continuous multivariate functions through the proposed operators.

Abstract

Abstract Here we expose multi-composite multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or R ^N, R N, N N N ∈ N, by the multi-composite multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by multi-composite iterated multi-layer operators of the last four types. These approximations are derived by establishing multi-composite multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multi-composite multidimensional density function induced by a multi-composite general sigmoid activation function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

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