A variation on the Stone-Weierstrass theorem

Jewett, Robert I.

doi:10.1090/s0002-9939-1963-0152882-6

Cited by 21 publications

(9 citation statements)

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“…A first application of the factorization described in this section: it leads at once from Theorem 2 of Jewett [6] to Theorem 2 of Prolla [7]. Our discussion from Section 4.4 will show that factorization is by itself an efficient tool for the study of density problems in functions spaces.…”

Section: Remarkmentioning

confidence: 94%

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Stone–Weierstrass theorems revisited

Timofte

2005

Journal of Approximation Theory

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We prove strengthened and unified forms of vector-valued versions of the Stone-Weierstrass theorem. This is possible by using an appropriate factorization of a topological space, instead of the traditional localizability. Our main Theorem 7 generalizes and unifies number of known results. Applications from the last section include new versions in the scalar case, as well as simultaneous approximation and interpolation under additional constraints.

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Section: Remarkmentioning

confidence: 94%

“…The set E is important because Stone-Weierstrass theorems typically state that various hypotheses imply equality in (6). …”

Section: Remarkmentioning

confidence: 99%

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Stone–Weierstrass theorems revisited

Timofte

2005

Journal of Approximation Theory

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“…The consideration of modules is essential for vector valued theorems, but even in the scalar case they are important since they allow one to describe the closure of vector subspaces and of convex cones which are not themselves algebras. [8] are used in the proof of Lemma 1 below. Notice that the proof of Lemma 1 is inspired by a result of Brosowski and Deutsch [3], as modified by Ransford [15].…”

Section: Convex Conesmentioning

confidence: 99%

A generalized Bernstein approximation theorem

Prolla

1988

Math. Proc. Camb. Phil. Soc.

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A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of fone for each integer n ≥ 1. Bernstein's Theorem states that Bn(f) → f uniformly on [0, 1] and, since each Bn(f) is a polynomial of degree at most n, we have as a consequence Weierstrass' theorem. See for example Lorentz [9]. The operator Bn, defined on the space C([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that Bn(f) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C+(X):= {f ∈ C(X; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C+(X) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C(X; ℝ), and, in particular, the closure of A+:= {f ∈ A; f ≥ 0}, where A is a subalgebra of C(X; ℝ).

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“…More precisely and based on ideas of Jewett [7] and Prolla [10], we provide a set of sufficient conditions on a subspace of the space of fuzzy-number-valued functions in order that it be dense, which is to say a Stone-Weierstrass type result. The celebrated Stone-Weierstrass theorem is one of the most important results in classical analysis, plays a key role in the development of general approximation theory and, particularly, is in the essence of the approximation capabilities of neural networks.…”

Section: Introductionmentioning

confidence: 99%

A version of the Stone-Weierstrass theorem in fuzzy analysis

Font¹,

Sanchis²,

Sanchis³

2017

J. Nonlinear Sci. Appl.

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Let C(K, E 1 ) be the space of continuous functions defined between a compact Hausdorff space K and the space of fuzzy numbers E 1 endowed with the supremum metric. We provide a set of sufficient conditions on a subspace of C(K, E 1 ) in order that it be dense. We also obtain a similar result for interpolating families of C(K, E 1 ). As a corollary of the above results we prove that certain fuzzy-number-valued neural networks can approximate any continuous fuzzy-number-valued function defined on a compact subspace of R n .

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A variation on the Stone-Weierstrass theorem

Cited by 21 publications

References 1 publication

Stone–Weierstrass theorems revisited

Stone–Weierstrass theorems revisited

A generalized Bernstein approximation theorem

A version of the Stone-Weierstrass theorem in fuzzy analysis

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