Separation conditions and criteria for uniform approximation of input-output maps
✍ Scribed by Sandberg, Irwin W.
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 108 KB
- Volume
- 26
- Category
- Article
- ISSN
- 0098-9886
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✦ Synopsis
We consider multidimensional shift-invariant input-output maps G from a relatively compact set of functions S to a set of real-valued functions, and we give criteria under which these maps can be uniformly approximated arbitrarily well using a certain structure consisting of a not-necessarily linear dynamic part followed by a non-linear memoryless section that may contain sigmoids or radial basis functions, etc. The dynamic part is comprised of a ÿnite number of dynamic maps h1; : : : ; hj drawn from a set H of maps that satisfy a certain continuity condition. In our results certain separation conditions, of the kind associated with the Stone-Weierstrass theorem, play a prominent role. Here they emerge as criteria for approximation, and not just su cient conditions under which an approximation exists. In particular, one of the theorems given is a result to the e ect that universal approximation can be achieved using the structure we consider if and only if the set H satisÿes the separation condition that (hu1)(0) = (hu2)(0) for some h ∈ H whenever u1; u2 ∈cl(S) and u1 = u2 (where cl(S) denotes the closure of S). This holds even if the elements of H are not linear. ?