Théorème de Stone-Weierstrass

Théorème de Stone-Weierstrass En analyse mathématique, le théorème d'approximation de Weierstrass stipule que toute fonction continue définie sur un intervalle fermé [un, b] peut être approché uniformément aussi étroitement que souhaité par une fonction polynomiale. Parce que les polynômes sont parmi les fonctions les plus simples, et parce que les ordinateurs peuvent évaluer directement les polynômes, ce théorème a une pertinence à la fois pratique et théorique, surtout en interpolation polynomiale. La version originale de ce résultat a été établie par Karl Weierstrass en 1885 en utilisant la transformée de Weierstrass.

Maréchal H. Stone a considérablement généralisé le théorème (Pierre 1937) et simplifié la preuve (Pierre 1948). Son résultat est connu sous le nom de théorème de Stone-Weierstrass. Le théorème de Stone-Weierstrass généralise le théorème d'approximation de Weierstrass dans deux directions: au lieu de l'intervalle réel [un, b], on considère un espace de Hausdorff arbitrairement compact X, et au lieu de l'algèbre des fonctions polynomiales, une variété d'autres familles de fonctions continues sur {style d'affichage X} sont montrés suffisants, comme détaillé ci-dessous. Le théorème de Stone-Weierstrass est un résultat essentiel dans l'étude de l'algèbre des fonctions continues sur un espace de Hausdorff compact.

Plus loin, il y a une généralisation du théorème de Stone – Weierstrass aux espaces de Tychonoff non compacts, à savoir, toute fonction continue sur un espace de Tychonoff est approchée uniformément sur des ensembles compacts par des algèbres du type apparaissant dans le théorème de Stone-Weierstrass et décrit ci-dessous.

Une généralisation différente du théorème original de Weierstrass est le théorème de Mergelyan, qui le généralise à des fonctions définies sur certains sous-ensembles du plan complexe.

Contenu 1 Théorème d'approximation de Weierstrass 1.1 Applications 2 Théorème de Stone-Weierstrass, version réelle 2.1 Version localement compacte 2.2 Applications 3 Théorème de Stone-Weierstrass, version complexe 4 Théorème de Stone-Weierstrass, version quaternion 5 Théorème de Stone-Weierstrass, Version C*-algèbre 6 Versions en treillis 7 Théorème de Bishop 8 Théorème de Nachbin 9 Histoire éditoriale 10 Voir également 11 Remarques 12 Références 12.1 Ouvrages historiques 13 External links Weierstrass approximation theorem The statement of the approximation theorem as originally discovered by Weierstrass is as follows: Weierstrass Approximation Theorem — Suppose  f  is a continuous real-valued function defined on the real interval [un, b]. For every ε > 0, il existe un polynôme p tel que pour tout x dans [un, b], Nous avons |F(X) −p(X)| < ε, or equivalently, the supremum norm || f  − p|| < ε. A constructive proof of this theorem using Bernstein polynomials is outlined on that page. Applications 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. Since C[a, b] is metrizable and separable it follows that C[a, b] has cardinality at most 2ℵ0. (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.) Stone–Weierstrass theorem, real version The set C[a, b] of continuous real-valued functions on [a, b], together with the supremum norm || f || = supa ≤ x ≤ b | f (x)|, is a Banach algebra, (that is, an associative algebra and a Banach space such that || fg|| ≤ || f ||·||g|| for all  f, g). The set of all polynomial functions forms a subalgebra of C[a, b] (that is, a vector subspace of C[a, b] that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a, b]. Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X, R) of real-valued continuous functions on X, with the topology of uniform convergence. He wants to find subalgebras of C(X, R) which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set A of functions defined on X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) ≠ p(y). Now we may state: Stone–Weierstrass Theorem (real numbers) — Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, R) which contains a non-zero constant function. Then A is dense in C(X, R) if and only if it separates points. This implies Weierstrass' original statement since the polynomials on [a, b] form a subalgebra of C[a, b] which contains the constants and separates points. Locally compact version A version of the Stone–Weierstrass theorem is also true when X is only locally compact. Let C0(X, R) be the space of real-valued continuous functions on X which vanish at infinity; that is, a continuous function  f  is in C0(X, R) if, for every ε > 0, il existe un ensemble compact K ⊂ X tel que| F | < ε on X K. Again, C0(X, R) is a Banach algebra with the supremum norm. A subalgebra A of C0(X, R) is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some  f  in A such that  f (x) ≠ 0. The theorem generalizes as follows: Stone–Weierstrass Theorem (locally compact spaces) — Suppose X is a locally compact Hausdorff space and A is a subalgebra of C0(X, R). Then A is dense in C0(X, R) (given the topology of uniform convergence) if and only if it separates points and vanishes nowhere. This version clearly implies the previous version in the case when X is compact, since in that case C0(X, R) = C(X, R). There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.[1] Applications The Stone–Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass's result. If  f  is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, alors il existe une fonction polynomiale p en deux variables telle que |F(X, y) −p(X, y)| < ε for all x in [a, b] and y in [c, d].[citation needed] If X and Y are two compact Hausdorff spaces and f : X × Y → R is a continuous function, then for every ε > 0 there exist n > 0 et fonctions continues  f1, ..., fn  sur X et fonctions continues g1, ..., gn sur Y tel que || f − Σ fi gi || < ε.[citation needed] The theorem has many other applications to analysis, including: Fourier series: The set of linear combinations of functions en(x) = e2πinx, n ∈ Z is dense in C([0, 1]/{0, 1}), where we identify the endpoints of the interval [0, 1] to obtain a circle. An important consequence of this is that the en are an orthonormal basis of the space L2([0, 1]) of square-integrable functions on [0, 1]. Stone–Weierstrass theorem, complex version Slightly more general is the following theorem, where we consider the algebra {displaystyle C(X,mathbb {C} )} of complex-valued continuous functions on the compact space {displaystyle X} , again with the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation. Stone–Weierstrass Theorem (complex numbers) — Let {displaystyle X} be a compact Hausdorff space and let {displaystyle S} be a separating subset of {displaystyle C(X,mathbb {C} )} . Then the complex unital *-algebra generated by {displaystyle S} is dense in {displaystyle C(X,mathbb {C} )} . The complex unital *-algebra generated by {displaystyle S} consists of all those functions that can be obtained from the elements of {displaystyle S} by throwing in the constant function 1 and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times. This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, {displaystyle f_{n}to f} , then the real parts of those functions uniformly approximate the real part of that function, {displaystyle operatorname {Re} f_{n}to operatorname {Re} f} , and because for real subsets, {displaystyle Ssubset C(X,mathbb {R} )subset C(X,mathbb {C} ),} taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated. As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces. Stone–Weierstrass theorem, quaternion version Following John C.Holladay (1957) : consider the algebra C(X, H) of quaternion-valued continuous functions on the compact space X, again with the topology of uniform convergence. If a quaternion q is written in the form q = a + ib;+ jc + kd then the scalar part a is the real number (q − iqi − jqj − kqk)/4. Likewise being the scalar part of −qi, −qj and −qk : b,c and d are respectively the real numbers (−qi − iq + jqk − kqj)/4, (−qj − iqk − jq + kqi)/4 and (−qk + iqj − jqk − kq)/4. Then we may state : Stone–Weierstrass Theorem (quaternion numbers) — Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, H) which contains a non-zero constant function. Then A is dense in C(X, H) if and only if it separates points. Stone–Weierstrass theorem, C*-algebra version The space of complex-valued continuous functions on a compact Hausdorff space {displaystyle X} i.e. {displaystyle C(X,mathbb {C} )} is the canonical example of a unital commutative C*-algebra {displaystyle {mathfrak {A}}} . The space X may be viewed as the space of pure states on {displaystyle {mathfrak {A}}} , with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which has remain unsolved, is as follows: Conjecture — If a unital C*-algebra {displaystyle {mathfrak {A}}} has a C*-subalgebra {displaystyle {mathfrak {B}}} which separates the pure states of {displaystyle {mathfrak {A}}} , then {displaystyle {mathfrak {A}}={mathfrak {B}}} . In 1960, Jim Glimm proved a weaker version of the above conjecture. Stone–Weierstrass theorem (C*-algebras)[2] — If a unital C*-algebra {displaystyle {mathfrak {A}}} has a C*-subalgebra {displaystyle {mathfrak {B}}} which separates the pure state space (i.e. the weak-* closure of the pure states) of {displaystyle {mathfrak {A}}} , then {displaystyle {mathfrak {A}}={mathfrak {B}}} . Lattice versions Let X be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in C(X, R). A subset L of C(X, R) is called a lattice if for any two elements  f, g ∈ L, the functions max{ f, g}, min{ f, g} also belong to L. The lattice version of the Stone–Weierstrass theorem states: Stone–Weierstrass Theorem (lattices) — Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R) with the property that for any two distinct elements x and y of X and any two real numbers a and b there exists an element  f  ∈ L with  f (x) = a and  f (y) = b. Then L is dense in C(X, R). The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value | f | which in turn can be approximated by polynomials in  f . A variant of the theorem applies to linear subspaces of C(X, R) closed under max (Hewitt & Stromberg 1965, Theorem 7.29): Stone–Weierstrass Theorem (max-closed) — Suppose X is a compact Hausdorff space and B is a family of functions in C(X, R) such that B separates points. B contains the constant function 1. If  f  ∈ B then af  ∈ B for all constants a ∈ R. If  f,  g ∈ B, then  f  + g, max{ f, g} ∈ B. Then B is dense in C(X, R). More precise information is available: Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R). The function φ ∈ C(X, R) belongs to the closure of L if and only if for each pair of distinct points x and y in X and for each ε > 0 il existe des  f  ∈ L pour lesquels |F(X) − f(X)| < ε and | f (y) − φ(y)| < ε. Bishop's theorem Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows (Bishop 1961): Bishop's theorem — Let A be a closed subalgebra of the complex Banach algebra C(X, C) of continuous complex-valued functions on a compact Hausdorff space X, using the supremum norm. For S ⊂ X we write AS = {g|S : g ∈ A}. Suppose that f  ∈ C(X, C) has the following property:  f |S ∈ AS for every maximal set S ⊂ X such that all real functions of AS are constant. Then  f  ∈ A. Glicksberg (1962) gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem : the process of Louis de Branges (1959). See also Rudin (1973, §5.7). Nachbin's theorem Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold (Nachbin 1949). Nachbin's theorem is as follows (Llavona 1986): Nachbin's theorem — Let A be a subalgebra of the algebra C∞(M) of smooth functions on a finite dimensional smooth manifold M. Suppose that A separates the points of M and also separates the tangent vectors of M: for each point m ∈ M and tangent vector v at the tangent space at m, there is a f ∈ A such that df(x)(v) ≠ 0. Then A is dense in C∞(M). Editorial history In 1885 it was also published an English version of the paper whose title was On the possibility of giving an analytic representation to an arbitrary function of real variable.[3][4][5][6][7] According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".[7][6] See also Müntz–Szász theorem Bernstein polynomial Runge's phenomenon shows that finding a polynomial P such that  f (x) = P(x) for some finely spaced x = xn is a bad way to attempt to find a polynomial approximating  f  uniformly. A better approach, explained e.g. in (Rudin 1976), p. 160, eq. (51) ff., is to construct polynomials P uniformly approximating  f  by taking the convolution of  f  with a family of suitably chosen polynomial kernels. Mergelyan's theorem, concerning polynomial approximations of complex functions. Notes ^ Willard, Stephen (1970). General Topology. Addison-Wesley. p. 293. ISBN 0-486-43479-6. ^ Glimm, James (1960). "A Stone–Weierstrass Theorem for C*-algebras". Annals of Mathematics. Second Series. 72 (2): 216–244 [Theorem 1]. doi:10.2307/1970133. JSTOR 1970133. ^ Pinkus, Allan. "Weierstrass and Approximation Theory" (PDF). Journal of Approximation Theory. 107 (1): 8. ISSN 0021-9045. OCLC 4638498762. Archived (PDF) from the original on October 19, 2013. Retrieved July 3, 2021. ^ Pinkus, Allan (2004). "Density methods and results in approximation theory". Orlicz Centenary Volume. Banach Center publications. Institute of Mathematics, Polish Academy of Sciences. 64: 3. CiteSeerX 10.1.1.62.520. ISSN 0137-6934. OCLC 200133324. Archived from the original on July 3, 2021. ^ Ciesielski, Zbigniew; Pełczyński, Aleksander; Skrzypczak, Leszek (2004). Orlicz centenary volume : proceedings of the conferences "The Wladyslaw Orlicz Centenary Conference" and Function Spaces VII : Poznan, 20-25 July 2003. Vol. I, Plenary lectures. Banach Center publications. Vol. 64. Institute of Mathematics. Polish Academy of Sciences. p. 175. OCLC 912348549. ^ Jump up to: a b Quintana, Yamilet; Perez D. (2008). "A survey on the Weierstrass approximation theorem". Divulgaciones Matematicas. 16 (1): 232. OCLC 810468303. Retrieved July 3, 2021. Weierstrass' perception on analytic functions was of functions that could berepresented by power series (arXiv 0611038v2). ^ Jump up to: a b Quintana, Yamilet (2010). "On Hilbert extensions of Weierstrass' theorem with weights". Journal of Function Spaces. Scientific Horizon. 8 (2): 202. doi:10.1155/2010/645369. ISSN 0972-6802. OCLC 7180746563. (arXiv 0611034v3). Citing: D. S. Lubinsky, Weierstrass' Theorem in the twentieth century: a selection, in Quaestiones Mathematicae18 (1995), 91–130. References John C.Holladay (1957), "The Stone–Weierstrass theorem for quaternions" (PDF), Proc. Amer. Math. Soc., 8: 656, doi:10.1090/S0002-9939-1957-0087047-7. Louis de Branges (1959), "The Stone–Weierstrass theorem", Proc. Amer. Math. Soc., 10 (5): 822–824, doi:10.1090/s0002-9939-1959-0113131-7. Jan Brinkhuis & Vladimir Tikhomirov (2005) Optimization: Insights and Applications, Princeton University Press ISBN 978-0-691-10287-0 MR2168305. Glimm, James (1960), "A Stone–Weierstrass Theorem for C*-algebras", Annals of Mathematics, Second Series, 72 (2): 216–244, doi:10.2307/1970133, JSTOR 1970133 Bishop, Errett (1961), "A generalization of the Stone–Weierstrass theorem", Pacific Journal of Mathematics, 11 (3): 777–783, doi:10.2140/pjm.1961.11.777. Glicksberg, Irving (1962), "Measures Orthogonal to Algebras and Sets of Antisymmetry", Transactions of the American Mathematical Society, 105 (3): 415–435, doi:10.2307/1993729, JSTOR 1993729. Hewitt, E; Stromberg, K (1965), Real and abstract analysis, Springer-Verlag. Rudin, Walter (1976), Principles of mathematical analysis (3rd. ed.), McGraw-Hill, ISBN 978-0-07-054235-8. Rudin, Walter (1973), Functional analysis, McGraw-Hill, ISBN 0-07-054236-8. Nachbin, L. (1949), "Sur les algèbres denses de fonctions diffèrentiables sur une variété", C. R. Acad. Sci. Paris, 228: 1549–1551 Llavona, José G. (1986), Approximation of continuously differentiable functions, Amsterdam: North-Holland, ISBN 9780080872414 JG Burkill, Lectures On Approximation By Polynomials (PDF). Historical works The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II). Erste Mitteilung (part 1) pp. 633–639, Zweite Mitteilung (part 2) pp. 789–805. Important historical works of Stone include: Stone, M. H. (1937), "Applications of the Theory of Boolean Rings to General Topology", Transactions of the American Mathematical Society, 41 (3): 375–481, doi:10.2307/1989788, JSTOR 1989788. Stone, M. H. (1948), "The Generalized Weierstrass Approximation Theorem", Mathematics Magazine, 21 (4): 167–184, doi:10.2307/3029750, JSTOR 3029750, MR 0027121; 21 (5), 237–254. External links "Stone–Weierstrass theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] hide Authority control National libraries France (data)GermanyIsraelUnited States Other SUDOC (France) 1 Categories: Continuous mappingsTheorems in analysisTheorems in approximation theory1885 in mathematics1937 in mathematics

Si vous voulez connaître d'autres articles similaires à Théorème de Stone-Weierstrass vous pouvez visiter la catégorie Continuous mappings.

Laisser un commentaire

Votre adresse email ne sera pas publiée.

Monter

Nous utilisons nos propres cookies et ceux de tiers pour améliorer l'expérience utilisateur Plus d'informations