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Topic: Weierstrass approximation theorem

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	PlanetMath: proof of Weierstrass approximation theorem in R^n
		It is a simple matter of rescaling variables to conclude the Weirestrass approximation theorem for arbitrary parallelopipeds.
		By approximating this extended function, we conclude the Weierstrass approximation theorem for arbitrary compact subsets of
		This is version 5 of proof of Weierstrass approximation theorem in R^n, born on 2006-02-06, modified 2006-02-07.
	www.planetmath.org /encyclopedia/ProofOfWeierstrassApproximationTheoremInRn.html (246 words)

	Stone-Weierstrass theorem
		In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.
		The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated.
		As a consequence of the Weierstrass approximation theorem, one can show that the space C[a,b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients.
	www.abacci.com /wikipedia/topic.aspx?cur_title=Stone-Weierstrass_theorem (937 words)

	NationMaster - Encyclopedia: Separable space (Site not responding. Last check: )
		In the same way that any real number can be approximated to any specified accuracy by rational numbers, a separable space has some countable subset with which all its elements can be approached, in the sense of a mathematical limit.
		The space of continuous functions on the unit interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (this is the Weierstrass approximation theorem).
		Since the space of continuous functions on the interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (see Weierstrass approximation theorem), and their coefficients can be approximated by rationals, that space is also separable.
	www.nationmaster.com /encyclopedia/Separable-space (2174 words)

	PlanetMath: proof of Weierstrass approximation theorem
		A corollary of the result just proven is the Weierstrass appriximation theorem for piecewise linear functions.
		Cross-references: endpoints, bounded, point, integer, uniformly continuous, compact, continuous, piecewise, triangle inequality, inequalities, approximation theorem, algebra, Weierstrass approximation theorem, converge, infinity, summing, sides, limit, implies, relation, monotonically increasing, definition, obvious, approximations, polynomial, Babylonian method of computing square roots, domain, maps, variable, interval, function
		This is version 5 of proof of Weierstrass approximation theorem, born on 2004-09-05, modified 2007-05-30.
	planetmath.org /encyclopedia/ProofOfWeierstrassApproximationTheorem.html (326 words)

	Stone-Weierstrass theorem - Encyclopedia, History, Geography and Biography
		Marshall H. Stone considerably generalized the theorem (Stone, 1937) and simplified the proof (Stone, 1948); his result is known as the Stone–Weierstrass theorem.
		The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated.
		Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below.
	www.arikah.com /encyclopedia/Stone-Weierstrass_theorem (1050 words)

	NationMaster - Encyclopedia: Polynomial interpolation (Site not responding. Last check: )
		Polynomials can be used to approximate more complicated curves, for example, the shapes of letters in typography, given a few points.
		The Bernstein form was used in a constructive proof of the Weierstrass approximation theorem by Bernstein and has nowadays gained great importance in computer graphics in the form of Bezier curves.
		In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point.
	www.nationmaster.com /encyclopedia/Polynomial-interpolation (2775 words)

	Stone-Weierstrass theorem
		The Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be approximated as closely as desired by a polynomial function.
		Marshall H. Stone[?] considerably generalized the theorem and simplified the proof; his result is known as the Stone-Weierstrass theorem.
		The approximation theorem is generalized in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from other subalgebras of C(X) is investigated.
	www.ebroadcast.com.au /lookup/encyclopedia/st/Stone-Weierstrass_theorem.html (502 words)

	[No title]
		Karl Weierstrass as Rector of the Humboldt-Universität in Berlin.
		Weierstrass' paper with his proof of the Weierstrass Theorem on density on algebraic polynomials in the space of continuous real-valued functions on any finite closed interval.
		This is the translation of the Weierstrass 1885 paper and, as the original, it appeared in two parts and in subsequent issues, but under the same title.
	www.math.technion.ac.il /hat/wei.html (198 words)

	PlanetMath: Weierstrass approximation theorem
		The Stone-Weierstrass theorem is a generalization to even more general situations.
		Cross-references: even, Stone-Weierstrass theorem, compact subsets, polynomial, interval, function, continuous
		This is version 4 of Weierstrass approximation theorem, born on 2004-09-05, modified 2006-03-01.
	www.planetmath.org /encyclopedia/WeierstrassApproximationTheorem.html (58 words)

	Interactive Mathematics Miscellany and Puzzles
		In contrast, the theorem in Chapter 18 is a classical result.
		The treatment of this theorem is based upon the Steiner transformation, which is smoothing.
		For instance, we obtain estimates for the distances between the control points and the curve; this yields the Weierstrass approximation theorem.
	www.cut-the-knot.org /books/over/preface.shtml (750 words)

	PETER LANG VERLAGSGRUPPE
		The Weierstrass-Stone Theorem is one of the main tools of modern analysis, and several parts of functional analysis would not exist without it.
		The purpose of this monograph is to present its true nature by proving several increasing generalizations of this theorem, going from the classical case of subalgebras to submodules and to arbitrary subsets of continuous functions over compact spaces.
		Some closely connected results on uniform approximation which are important for many applications are also presented, namely the Choquet-Deny and the Kakutani Theorems for semi-lattices and for lattices of continuous functions, respectively.
	www.peterlang.com /index.cfm?vID=46511&vLang=D&vHR=1&vUR=2&vUUR=5 (254 words)

	Stone-Weierstrass theorem - Glasgledius (Site not responding. Last check: )
		The Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be approximated as closely as desired by a polynomial function.
		Marshall H. Stone[?] considerably generalized the theorem and simplified the proof; his result is known as the Stone-Weierstrass theorem.
		Note that the above theorem is also true if we replace the assumption that A separate points with the slightly weaker assumption that for every two different points x and y in X there exists a function p in A with p(x) not equal to p(y).
	www.glasglow.com /E2/we/Weierstrass_approximation_theorem.html (502 words)

	Brouwer fixed point theorem Summary
		Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, bounded, connected, without holes, etcetera).
		The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis.
		The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of D
	www.bookrags.com /Brouwer_fixed_point_theorem (2012 words)

	American Mathematical Monthly - January 1997
		From this unconventional definition, an n-dimensional analogue of the "polar coordinates change of variable theorem" follows with relative ease.
		Mixing in some well-cured properties of Euler's gamma function, we compute the integral of a monomial over S^n-1 and thereby deduce the divergence theorem for polynomial fields over the unit ball in IR^n.
		The Weierstrass approximation theorem is then used to complete the proof of the divergence theorem for balls in IR^n.
	www.maa.org /pubs/monthly_jan97_toc.html (371 words)

	Introduction to Topology and Modern Analysis
		Recognizing that the only functions able to be continuous on a space with the indiscrete topology are the constants, and that a space with the discrete topology has continuous functions in abundance, the author asks the reader to consider topologies that fall between these extremes, and this motivates the separation properties of topological spaces.
		These theorems guarantee that the study of Banach spaces is worth doing, and that there are analogs of the finite dimensional theory in the (infinite)-dimensional context of Banach spaces.
		Indeed, the Gelfand-Naimark theorem, that essentially states that elements of a commutative Banach *-algebra act like the functions on its maximal ideal space, has to rank as one of the most interesting results in the book, and indeed in all of mathematics.
	www.8notes.com /books/detpage.asp?asin=1575242389&field-keywords=Couperin&schMod=music&type=&sb=s (1264 words)

	Stone-Weierstrass theorem - Art History Online Reference and Guide
		Marshall H. Stone considerably generalized the theorem (Stone, 1937) and simplified the proof (Stone, 1948); his result is known as the Stone-Weierstrass theorem.
		The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated.
		Further, there is a generalization of the Stone-Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space space is approximated uniformly on compact sets by algebras of the type appearing in the Stone-Weierstrass theorem and described below.
	www.arthistoryclub.com /art_history/Weierstrass_approximation_theorem (965 words)

	Content Guide
		Chapter 15 What is Slope (poster material)provides motivation for the standard approximation for the slope (or derivative) of a nonlinear function and then says if the approximation converge to a limit, that limit is the slope (or derivative).
		Chapter 17 What is Area (poster material) provides motivation for the standard approximation for the area of a region and then says if the approximation converges to a limit, that limit is the area.
		The assumption that the limit of area approximations exists is justified by the first fundamental theorem of calculus.
	whyslopes.com /etc/Real-Analysis-Decimal-View (1769 words)

	Citations: Fuzzy systems as universal approximators - Kosko (ResearchIndex) (Site not responding. Last check: )
		, that fuzzy controllers are universal approximators in the sense that it is possible to construct such rule bases that approximate uniformly any continuous function de ned on a compact subset of R with arbitrary accuracy.
		....theorems 3 and 4 imply that the stabilized KH controllers are capable of approximating any continuous function on a compact domain with respect to the supremum or L p (p norm.
		Let us look at the definition of a fuzzy logic device (FLD) The FLD is a general concept in which a deterministic output (crisp values) is the result of the mapping of deterministic inputs, starting from a set of rules relating linguistic variables to one another using fuzzy logic.
	citeseer.ist.psu.edu /context/112575/0 (2174 words)

	IMACS (Site not responding. Last check: )
		Groups and subgroups; symmetric groups; cycle notation; Lagrange's Theorem; permutation representations of groups; generators and generated subgroups; the Sylow Theorems; a classification of all finite Abelian groups; normal subgroups; quotient groups; the Isomorphism Theorems; simple groups; solvable groups; the Jordan-Hlder Theorem.
		Differentiability; the Linear Approximation Theorem; properties of derivatives; the calculus within its historical context; the Mean Value Theorem for Derivatives; curve sketching; the chain rule; parametric representation of relations; various forms of l'Hpital's rule; Cauchy's Mean Value Theorem; implicit differentiation; antiderivatives.
		The axiom of choice; the Hausdorff Chain Theorem; Zorn's lemma; the Well-Ordering Theorem; the principle of transfinite induction; Bourbaki's Theorem; transfinite recursion; ordinal numbers; cardinal numbers; a discussion of the continuum hypothesis; the Fundamental Theorem of Cardinal Arithmetic.
	imacs.org /IMACSWeb/default.aspx?page=Mathematics (1391 words)

	Continuous Function (Site not responding. Last check: )
		The following applet shows the progress of successive Bernstein polynomials in approximating a continuous function.
		You may select one of the three predefined functions, or you may sketch a function by dragging the mouse from left to right over the plotting region.
		Theorem 2.1 Assume g is a continuous function and pn from n=0 to n=infinity is a sequence determined by a fixed point iteration.
	www.lycos.com /info/continuous-function.html (384 words)

	The Weierstrass Approximation Theorem
		This is the conclusion of the famous Weierstrass approximation theorem, named for Karl Weierstrass.
		The proof of the Weierstrass theorem by Sergi Bernstein is constructive: it defines explicitly a sequence of polynomials that converge to
		Proofs of both theorems may also be found in most books on numerical analysis or approximation theory, for example, Isaacson and Keller (1994), Rivlin (1969), and Timan (1994).
	www.joma.org /images/upload_library/4/vol6/Mayans/Weierstrass.xml (276 words)

	The Weierstrass approximation theorem
		One of the most important ways in which a metric is used is in approximation.
		We are told that f is continuous and by a theorem we will prove in the next section we may assume that f is bounded by M (say).
		The Weierstrass theorem is about functions which are continuous on a closed bounded inteval like [a, b].
	www-groups.mcs.st-andrews.ac.uk /~john/analysis/Lectures/L19.html (552 words)

	Weierstrass Approximation Theorem
		The Weierstrass approximation theorem assures us that polynomial approximation can get arbitrarily close to any continuous function as the polynomial order is increased.
		Thus, any continuous function can be approximated arbitrarily well by means of a polynomial.
		The main point here is that, thanks to the Weierstrass approximation theorem, we know that good polynomial approximations exist for any continuous function.
	ccrma-www.stanford.edu /%7ejos/mdft/Weierstrass_Approximation_Theorem.html (168 words)

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