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Here we study the quantitative multivariate approximation of perturbed hyperbolic tangent activated singular integral operators to the unit operator. The engaged neural network activation function is both parametrized and deformed and the related kernel is a density function on \(\mathbb {R}^{N}\) . We exhibit uniform and \(L_{p}\) , \(p\ge 1\) , approximations via Jackson type inequalities involving the first \(L_{p}\) modulus of smoothness, \(1\le p\le \infty \) . Differentiability of our multivariate functions is covered extensively in our approximations.

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Perturbed Hyperbolic Tangent Activated Singular Integrals Multivariate Approximation with Rates

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  • 发表日期：

    2026年

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  • George A. Anastassiou

摘要

Here we study the quantitative multivariate approximation of perturbed hyperbolic tangent activated singular integral operators to the unit operator. The engaged neural network activation function is both parametrized and deformed and the related kernel is a density function on \(\mathbb {R}^{N}\) . We exhibit uniform and \(L_{p}\) , \(p\ge 1\) , approximations via Jackson type inequalities involving the first \(L_{p}\) modulus of smoothness, \(1\le p\le \infty \) . Differentiability of our multivariate functions is covered extensively in our approximations.