Stone-Weierstrass Research Articles

The HOL Light Theory of Euclidean Space

We describe the library of theorems about N-dimensional Euclidean space that has been formalized in the HOL Light prover. This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery. The library includes various `big name' theorems (Brouwer's fixed point theorem, the Stone-Weierstrass theorem, the Tietze extension theorem), numerous non-trivial results that are useful in applications (second mean value theorem for integrals, power series for real and complex transcendental functions) and a host of supporting definitions and lemmas. It also includes some specialized automated proof tools. The library has as planned been applied to the Flyspeck project and has become the basis of a significant development of results in complex analysis, among others.

Finite rank Bargmann–Toeplitz operators with non-compactly supported symbols

Theorems about characterization of finite rank Toeplitz operators in Fock–Segal–Bargmann spaces, known previously only for symbols with compact support, are carried over to symbols without that restriction, however with a rather rapid decay at infinity. The proof is based upon a new version of the Stone–Weierstrass approximation theorem.

Noncommutative homogeneous spaces: The matrix case

Given a quantum subgroup G⊂Un and a number k≤n we can form the homogeneous space X=G/(G∩Uk), and it follows from the Stone–Weierstrass theorem that C(X) is the algebra generated by the last n−k rows of coordinates on G. In the quantum group case the analogue of this basic result does not necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the “easy quantum group” case, with the construction and study of several algebras associated to the noncommutative spaces of type X=G/(G∩Uk+).

POLYNOMIAL APPROXIMATION, LOCAL POLYNOMIAL CONVEXITY, AND DEGENERATE CR SINGULARITIES — II

We provide some conditions for the graph of a Hölder-continuous function on [Formula: see text], where [Formula: see text] is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on [Formula: see text]. These conditions often fail to yield any conclusion if rank ℝDF is not maximal on a sufficiently large subset of [Formula: see text]. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ2 at an isolated complex tangency.

A Spectral Condition for Strong Compactness

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. A spectral condition is provided for an operator to be strongly compact and for its commutant to be a strongly compact algebra. This condition is applied to test strong compactness for some classes of operators, namely, bilateral weighted shifts, Cesaro operators, and composition operators. Some tools from function theory like the Mellin transform, the principle of analytic continuation, and the Weierstrass approximation theorem arise naturally in the pursuit of this program.

On the conservatism of the sum-of-squares method for analysis of time-delayed systems

In this paper, we present a converse Lyapunov result which shows that using polynomial Lyapunov functionals to prove the stability of linear time-delay systems is not conservative. This result motivates the sum-of-squares approach to stability analysis of linear time-delay systems. This main result is based on an extension of the Weierstrass approximation theorem to show that linear varieties of polynomials can be used to approximate linear varieties of the space of continuous functions.

Weierstrass’s theorem — Leaving no ‘Stone’ unturned

In this article, we discuss the basic theme of approximating functions by polynomial functions. Although it is exemplified by the classical theorem of Weierstrass, the theme goes much further. Even on the face of it, the advantage of polynomial approximations can be seen from the fact that unlike general continuous functions, it is possible to numerically feed polynomial interpolations of such functions into a computer and the justification that we will be as accurate as we want is provided by the theorems we discuss. In reality, this theme goes deep into subjects like Fourier series and has applications like separability of the space of continuous functions. Marshall Stone’s generalisation to compact Hausdorff spaces is natural and important in mathematics. Applications of the Weierstrass approximation theorem abound in mathematics — to Gaussian quadrature for instance.

Interval type-2 fuzzy neural networks version of the Stone–Weierstrass theorem

The main purpose of this study is to state conditions that guarantee an interval type-2 triangular fuzzy (IT2TF) neural network can approximate continuous IT2TF functions. To make a more efficient calculation with IT2TF numbers, the sum and the product of two IT2TF numbers are constructed. These concepts are used in the definition of IT2TF polynomials. Moreover, the present study provides a mathematical framework to show that IT2TF polynomials are a compact Hausdroff space. Based on this concept we establish an interval type-2 fuzzy neural networks version of the Stone–Weierstrass theorem which enables approximation by a special class of IT2TF neural networks on the set of all monotonic and continuous IT2TF functions. Finally, a numerical example is given to illustrate the results.

New representation of the remainder in the Bernstein approximation

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.

An Efficient Combinatorial-Probabilistic Dual-Fusion Modification of Bernstein’s Polynomial Approximation Operator

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 the polynomials which approximate the given function . An efficient ‘Combinatorial-Probabilistic Dual-Fusion’ version of the modification of Bernstein’s Polynomial Operator is proposed. The potential of the aforesaid improvement 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.

Strict topologies on spaces of vector-valued functions

Let X be a completely regular Hausdorff space, and {Ex : x ∈ X} a collection of non-trivial locally convex topological vector spaces indexed by X. Let E be their disjoint union. We investigate a species of strict topology on a vector space F of choice functions σ : X → E (σ(x) ∈ Ex), and obtain Stone–Weierstrass and spectral synthesis analogues. We also obtain completeness results in some special cases.

TRUNCATED SIMPLICIAL RESOLUTIONS AND SPACES OF RATIONAL MAPS

We show that whenever m ≤ n, the space of all continuous rational maps fromCP m toCP n has the same homology as the space of all continuous maps between these spaces in dimensions smaller than d(2n − 2m + 1) − 1. This improves the result of the paper ‘Spaces of rational maps and the Stone–Weierstrass theorem’ (Topology 45 (2006), 281–293).

Some approximation properties of adaptive fuzzy systems with variable universe of discourse

In the last 20 years, while most research on fuzzy approximation theory has focused on nonadaptive fuzzy systems, little work has been done on adaptive fuzzy systems. This paper introduces an algorithm for adaptive fuzzy systems with Variable Universe of Discourse (VUD). By means of contraction–expansion factors, universe of discourse can be modified online, and fuzzy rules can reproduce automatically to adapt to the modified universe of discourse. Thus, dependence on the size of initial rule base is greatly reduced. Using Stone–Weierstrass theorem, VUD adaptive fuzzy systems are proved to be universal approximators with two-order approximation accuracy. In addition, the convergence properties of approximation error are discussed, and a sufficient condition is presented to partition universe of discourse and to calculate the size of rule base. An example is also given to illustrate the approximation power of VUD adaptive fuzzy systems.

LINEAR MOTION BLUR IDENTIFICATION IN NOISY IMAGES USING BISPECTRUM AND FEED-FORWARD BACK PROPAGATION NEURAL NETWORKS

Motion blur is one of the most common causes of image corruptions caused by blurring. Several methods have been presented up to now, which precisely identify linear motion blur parameters, but most of them possessed low precision in the presence of the noise. The present paper is aimed to introduce an algorithm for estimating linear motion blur parameters in noisy images. This study presents a method to estimate motion direction by using Radon transform, which is followed by the application of two other different methods to estimate motion length; the first of which is based on one-dimensional power spectrum to estimate parameters of noise free images and the second uses bispectrum modeling in noisy images. A Feed-Forward Back Propagation neural network has been designed on the basis of Weierstrass approximation theorem to model bispectrum and the Delta rule as the network learning rule. The methods were tested on several standard images like Camera man, Lena, Lake, etc. that were degraded by linear motion blur and additive noise. The experimental results have been satisfactory. The proposed method, compared to other related methods, suggests an improvement in the supported lowest SNR and precision of estimation.

On Hilbert extensions of Weierstrass′ theorem with weights

In this paper we study the set of ℊ‐valued functions which can be approximated by ℊ‐valued continuous functions in the norm , where I ⊂ ℝ is a compact interval, ℊ is a separable real Hilbert space and w is a certain ℊ‐valued weakly measurable weight. Thus, we obtain a new extension of the celebrated Weierstrass approximation theorem.

Exponentially Stable Nonlinear Systems Have Polynomial Lyapunov Functions on Bounded Regions

This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set. The main result states that if there exists an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of Rn, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the vector field is at least <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, polynomials can be used to approximate a differentiable function, using the Sobolev norm <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,infin</sup> to any desired accuracy. This approximation result is combined with the second-order Taylor series expansion to show that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. The investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.

Ordered spaces, metric preimages, and function algebras

We consider the Complex Stone–Weierstrass Property (CSWP), which is the complex version of the Stone–Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results about those compact Hausdorff spaces which have scattered-to-one maps onto compact metric spaces.

Generalization of the B *‐algebra (C0(X),∥ ∥∞)

AbstractWe give a generalization of the Stone–Weierstrass property for subalgebras of C (X), with X a completely regular Hausdorff space. In particular, we study in this paper some subalgebras of C0(X), with X a locally compact Hausdorff space, provided with weighted norm topology. By using the Stone–Weierstrass property, we then describe the ideal structure of these algebras. (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A Stone–Weierstrass theorem for Banach function spaces satisfying a certain separation property

We consider a strong lattice property for a Banach function space B on a compact Hausdorff space, which gives a general Stone–Weierstrass theorem for B. We also study the relation of this theorem and its proof to a certain decomposition of an associated compactification, and to another lattice-like property.

Effectivity on Continuous Functions in Topological Spaces

In this paper we investigate aspects of effectivity and computability on partial continuous functions in topological spaces. We use the framework of TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. We generalize the representations introduced in [Weihrauch, K., “Computable Analysis,” Springer, Berlin, 2000] for the Euclidean case to computable T0-spaces and computably locally compact Hausdorff spaces respectively. We show their equivalence and in particular, prove an effective version of the Stone-Weierstrass approximation theorem.