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Corrected incorrect use of plural form of "criterion". Tags: Visual edit Mobile edit Mobile web edit |
→Arbitrary-width case: Even though Cybenko affirms he uses the "Riesz Representation theorem" (which applies only to Hilbert spaces) in his 1989 paper, he actually uses its generalized version, the Riesz–Markov–Kakutani representation theorem-- which applies to the Banach spaces he considers. |
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The case where <math>\sigma</math> is a generic non-polynomial function is harder, and the reader is directed to.<ref name="pinkus" />}}
The above proof has not specified how one might use a ramp function to approximate arbitrary functions in <math>C_0(\R^n, \R)</math>. A sketch of the proof is that one can first construct flat bump functions, intersect them to obtain spherical bump functions that approximate the [[Dirac delta function]], then use those to approximate arbitrary functions in <math>C_0(\R^n, \R)</math>.<ref>{{Cite journal |last=Nielsen |first=Michael A. |date=2015 |title=Neural Networks and Deep Learning |url=http://neuralnetworksanddeeplearning.com/ |language=en}}</ref> The original proofs, such as the one by Cybenko, use methods from functional analysis, including the [[Hahn–Banach theorem|Hahn-Banach]] and [[
Notice also that the neural network is only required to approximate within a compact set <math>K</math>. The proof does not describe how the function would be extrapolated outside of the region.
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