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IEEE Transactions on Neural Networks
|February 6, 2008
Summary
This paper refutes claims that sigmoidal function boundedness is irrelevant to approximation theorems. We demonstrate that boundedness is indeed a necessary condition for the validity of our approximation theorem.
Area of Science:
- Mathematical analysis
- Approximation theory
Background:
- The validity of an approximation theorem was previously discussed.
- A comments letter questioned the necessity and sufficiency of sigmoidal function boundedness.
Discussion:
- This paper directly addresses and refutes the claims made by Huang et al.
- We provide a rigorous mathematical argument to counter their assertions regarding sigmoidal function properties.
Key Insights:
- The boundedness of a sigmoidal function is a necessary condition for the approximation theorem's validity.
- The claim that boundedness is neither sufficient nor necessary is shown to be incorrect.
Outlook:
- This clarification reinforces the foundational principles of the original approximation theorem.
- Further research may explore the precise conditions for various approximation theorems.
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