Stone-Weierstrass theorem


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Stone-Weierstrass theorem

[′stōn ′vī·ər‚sträs ‚thir·əm]
(mathematics)
If S is a collection of continuous real-valued functions on a compact space E, which contains the constant functions, and if for any pair of distinct points x and y in E there is a function ƒ in S such that ƒ(x) is not equal to ƒ(y), then for any continuous real-valued function g on E there is a sequence of functions, each of which can be expressed as a polynomial in the functions of S with real coefficients, that converges uniformly to g.
References in periodicals archive ?
Recently Liaqat Ali Khan et al [6], has given a short proof of a weaker form of the Stone-Weierstrass Theorem for C(X).
By the Stone-Weierstrass theorem [B.sub.1] [R] C([Y.sub.1]) is uniformly dense in C([X.sub.1] x [Y.sub.1]); hence any element in [??] is uniformly approximated by [cross product] [R] C([Y.sub.1]).
We use the following Stone-Weierstrass theorem to prove the theorem.
Applying Stone-Weierstrass Theorem to the IT2FNN-0 Architecture.
Applying the Stone-Weierstrass Theorem to the IT2FNN-2 Architecture.
By using the Stone-Weierstrass theorem together with Lemmas 11 and 12, we establish that the proposed IT2FNN-2 possesses the universal approximation capability.
Therefore by choosing appropriate class of interval type-2 membership functions, we can conclude that the IT2FNN-0 and IT2FNN-2 with simplified fuzzy if-then rules satisfy the five criteria of the Stone-Weierstrass theorem.
In these experiments, the estimated RMSE values for nonlinear function identification with 10-fold cross-validation for the hybrid architectures (IT2FNN-2:A2C0 and IT2FNN-2: A2C1) illustrate the proof based on Stone-Weierstrass theorem, that they are universal approximators for efficient identification of nonlinear functions, complying with [absolute value of (g(x) - f(x))] < [epsilon].
The non-commutative version of the Stone-Weierstrass theorem (which is an open problem) says that: does there exist a proper rich [sup.*]-subalgebra in a given [C.sup.*]-algebra A?