Stone-Weierstrass Research Articles

Stone-Weierstrass type theorems for large deviations

We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $\Lambda(\cdot)_{\mid \mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $\Lambda(\cdot)_{\mid \mathcal{A}}$.

An extension of the weierstrass approximation theorem to linear varieties: application to delay systems

In this paper, we show that linear varieties of polynomials can be used to approximate linear varieties of the space of continuous functions. This property is important in applications where polynomial optimization is used as it allows one to impose affine constraints on the decision variables with no loss of accuracy. In particular, construction of Lyapunov functionals for systems with delay is discussed.

On the Stone and Bishop appoximation theorems

Generalizations of the Stone-Weierstrass and Bishop approximation theorems are presented. Given an algebra, a subspace in a continuous function space coinciding with the closure of this algebra is constructed. Analogs of these results are obtained in the case where the set of functions under consideration is not an algebra, but its closure is related to some algebra.

Nonlinear Multifunctional Sensor Signal Reconstruction Based on Total Least Squares

The least squares method is often used to estimate the parameters in multi-functional sensor signal reconstruction. If the data has been contaminated, the computational result of the method turns out to be insignificant. Two methods presented in this paper are suitable for different nonlinear conditions, which are based on the combination of the total least squares algorithm with the local linearization strategy and Stone-Weierstrass theorem. The two methods evaluate both the sensor output bias and its input error. The results of emulation and theory analysis indicate that the proposed algorithms are more accurate and reliable for signal reconstruction.

On Müntz Theorem for countable compact sets

In this note we solve the Muntz problem for the space C(K) whenever K (0,1) is a countable compact set which satisfies certain additional as- sumptions and we propose the general case as an open question. Every introductory course in Approximation Theory should contain at least one proof of the following classical result, which is a nice generalization of the Weierstrass approximation theorem: Theorem 1 (Muntz, 1914). Let

Isometries of the Dirichlet space among the composition operators

We show that every composition operator which is an isometry of the Dirichlet space is induced by a univalent full map of the disk into itself that fixes the origin. This is an analogue of the Hardy space result for inner functions due to Nordgren. The proof relies on the Stone-Weierstrass theorem and the Riesz representation theorem.

A Choquet-Deny-Type Theorem and Applications to Approximation in Weighted Spaces

We present a Choquet-Deny-type theorem in weighted spaces together with an application to simultaneous approximation and interpolation from inf-lattices generated by convex sets. Moreover, we determine a characterization of the Korovkin closure of vector lattices and, as a consequence, a different proof of the Stone-Weierstrass theorem for some weighted spaces.

Spaces of rational maps and the Stone–Weierstrass theorem

It is shown that Segal's theorem on the spaces of rational maps from CP 1 to CP n can be extended to the spaces of continuous rational maps from CP m to CP n for any m ⩽ n . The tools are the Stone–Weierstrass theorem and Vassiliev's machinery of simplicial resolutions.

Stone--Weierstrass theorems revisited

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 tra...

Stone–Weierstrass theorems revisited

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.

Complex function algebras and removable spaces

The compact Hausdorff space X has the Complex Stone–Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone–Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.

Twentieth-Century Gems from Mathematics Magazine

Would we turn to MATHEMATICS MAGAZINE today for a seminal research paper in analysis by one of the current giants of mathematics? Perhaps not. It would have been a good idea in 1948, though, when Marshall H. Stone published in the MAGAZINE [10] what came to be known as the general Stone-Weierstrass theorem. Though this may be the most striking example of a new and important result to appear first in MATHEMATICS MAGAZINE, there are many other examples of interesting articles written for the MAGAZINE by eminent mathematicians. The MAGAZINE has also been a rich resource of gems by less well-known mathematicians, and its backlog can be mined for all sorts of purposes. In the following pages we'll sample some of the items that we found attractive and interesting. They are highly personal choices, of course. The emphasis will be on the earlier years of the MAGAZINE, in part because many readers will be familiar with more recent material. In addition, the first author was Editor of the MAGAZINE from 1986 to 1990 and finds it too difficult to choose

Korovkin and Weierstrass Approximation via Lacunary Statistical Sequences

In this study we shall extended Korovkin and Weierstrass approximation theorem to lacunary statistical convergent sequences. In addition, to these approximation theorems, we established also introduced lacunary statistically convergent of degreeand establish a corresponding Korovkin type theorem namely the following: If the sequence of positive linear operators Pn: CM (a, b) → B(a, b) satisfies the conditions: * ||Pn(1, x)-1||� → 0(S 1 β θ ) as r → ∞ , * ||Pn(t, x)-x||B → 0(S 2 β θ ) as r → ∞ and * ||Pn(t 2 , x)-x 2 ||B → 0(S 3 β θ ) as r → ∞ , then for any function f ∈ CM (a, b), we have ||Pn (f, x)- (x)||B → 0(S β θ ) as r → ∞ and � =min{�1, � 2, � 3}.

Dominated Convergence and Stone-Weierstrass Theorem

Let C(X;R) the algebra of continuous real valued functions defined on a locally compact space X. We consider linear subspaces A ⊂ C(X;R) having the following property: there is a sequence (Φj)j∈N of positive functions in A with limx→∞ Φj(x) = +∞ for every j ∈ N, such that A consists of functions f ∈ C(X;R) bounded above for the absolute value by an homothetic of some Φn (n depends on each f). Dominated convergence of a sequence (gn)n≥1 in A is an estimation of the form |gn(x) − g(x)| ≤ en|h(x)| for all x ∈ X and all n ∈ N where gn, g, h ∈ A and en → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices B ⊂ A and we show that the dominated convergence for sequences is exactly the convergence of sequences when A is endowed with a locally convex (DF)-space topology.

The Weierstrass theorem on polynomial approximation

In the paper a simple proof of the Weierstrass approximation theorem on a function continuous on a compact interval of the real line is given. The proof is elementary in the sense that it does not use uniform continuity.

AVERAGE LINEAR APPROXIMATIONS FOR SMOOTH FUNCTIONS

Average linear approximations for smooth functions using empirical and Gaussian probabilities are introduced. The convergence of these approximations is shown to be uniform, in the sense of the Weierstrass approximation theorem. Average linear approximations with prearranged error functions are finally studied.

Stone-weierstrass theorems for group-valued functions

Constructive groups were introduced by Sternfeld in [6] as a class of metrizable groupsG for which a suitable version of the Stone-Weierstrass theorem on the group ofG-valued functionsC(X, G) remains valid. As a way of exploring the existence of such Stone-Weierstrass-type theorems in this context we address the question raised in [6] as to which groups are constructive and prove that a locally compact group with more than two elements is constructive if and only if it is either totally disconnected or homeomorphic to some vector group ℝn. It may therefore be concluded that the Stone-Weierstrass theorem can be extended to some noncommutative Lie groups — exactly to those not containing any nontrivial compact subgroup.

An extended urn model with application to approximation

Pólya’s urn model from probability theory is extended to obtain a class of approximation operators for which the Weierstrass Approximation Theorem holds.

Equivalence of the Stone-Weierstrass conjectures for C* and JB*-algebras

We prove that the Stone-Weierstrass conjectures for C*-algebras and JB*-algebras are equivalent.

An iterative algorithm for improved approximation by Bernstein’s operator using statistical perspective

The celebrated Weierstrass Approximation Theorem (1885) heralded intermittent interest in polynomial approximation, which continues unabated even as of today. The great Russian mathematician Bernstein in 1912 not only provided an interesting proof of the Weierstrass’ theorem, but also displayed a sequence of polynomials that approximate a given function f∈ C[0,1]. This paper dwells upon constructing an iterative algorithm with an intention of improving the degree of approximation by this Bernstein’s polynomial. The perspective motivating the proposed algorithm is Statistical, and works through a fuller use of the information, about the unknown function f being approximated, which is available in terms of its known values at equidistant knots in C[0,1]. This information is used aposteriori to improve upon the approximation by the Bernstein polynomial. This improvement turns out to be available iteratively, which is seminal to the iterative nature of the proposed algorithm of the improvement of the degree of approximation by the Bernstein polynomial. The potential of the aforesaid improvement algorithm is tried to be brought forth and illustrated through an empirical study for which the function is assumed to be known in the sense of simulation.