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In this chapter we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.

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Uniform Approximation with Rates by Activated Singular Operators

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  • 发表日期：

    2026年

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  • George A. Anastassiou

摘要

In this chapter we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.