Stone-Weierstrass theorem

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The approximation theorem by Stone-Weierstrass (after Marshall Harvey Stone and Karl Weierstrass ) is a theorem from analysis that says under which conditions every continuous function can be approximated arbitrarily well by simpler functions.

sentence

Each sub-algebra P of the function algebra  A of the continuous real-valued or complex-valued functions on a compact Hausdorff space  M ,

  • the points separating is ,
  • for none of their evaluation functions , the zero function is ,
  • and - in the event that the base body of the field of complex numbers is - with respect to complex conjugation is complete, the thus with each also the associated complex conjugate function in P included is

is close to A with respect to the topology of uniform convergence .

That means: Every continuous function of M in the basic field can be approximated equally well as desired by functions from P under the given conditions .

Inferences

  • This theorem is a generalization of Weierstrass's approximation theorem, according to which every continuous function can be approximated uniformly on a compact interval using polynomials. This special case can easily be derived from the above general theorem if one takes the set of polynomials as sub-algebra P (see also Bernstein polynomials ).
  • Another important conclusion (often also referred to as Weierstraß's approximation theorem) is that every continuous function on the compact interval [0,2π] with the same value at 0 and 2π is uniformly represented by trigonometric polynomials (i.e. polynomials in sin ( x ) and cos ( x ) or linear combinations of sin ( nx ) and cos ( nx ), n ∈ℕ) can be approximated (see also the article on Fourier series ).
  • By means of the Alexandroff compactification , the sentence is also transferred to the space of the functions (see there) on a locally compact Hausdorff space .

history

In 1885 Weierstrass published a proof of his theorem. Independently of this, several mathematicians found further evidence, e.g. Runge (1885), Picard (1891), Volterra (1897), Lebesgue (1898), Mittag-Leffler (1900), Fejér (1900), Lerch (1903), Landau (1908) , de La Vallée Poussin (1912) and Bernstein (1912).

Generalizations

Several generalizations were found for Stone-Weierstrass's approximation theorem, such as Bishop's theorem . The lemma of Machado is closely connected to both theorems , with the help of which a generalized version of the approximation theorem by Stone-Weierstrass can be derived, which extends it to any Hausdorff spaces and the associated function algebras of the infinitely vanishing continuous functions .

literature

  • Kurt Endl, Wolfgang Luh : Analysis II . Aula-Verlag 1972. 7th edition. 1989, ISBN 3-89104-455-0 , pp. 132-134
  • Lutz Führer: General topology with applications . Vieweg Verlag, Braunschweig 1977, ISBN 3-528-03059-3 .
  • Jürgen Heine: Topology and Functional Analysis . Basics of abstract analysis with applications. 2nd, improved edition. Oldenbourg Verlag, Munich 2011, ISBN 978-3-486-70530-0 .
  • Friedrich Hirzebruch , Winfried Scharlau : Introduction to Functional Analysis (=  series "BI University Pocket Books" . Volume 296 ). Bibliographical Institute, Mannheim / Vienna / Zurich 1971, ISBN 3-411-00296-4 ( MR0463864 ).
  • Konrad Köngisberger: Analysis 1 . 2nd Edition. Springer 1992, ISBN 3-540-55116-6 , pp. 302-304
  • Walter Rudin : Functional Analysis (=  International Series in Pure and Applied Mathematics ). 2nd Edition. McGraw-Hill, Boston (et al.) 1991, ISBN 0-07-054236-8 ( MR1157815 ).
  • Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
  • Mícheál Ó Searcóid: Elements of Abstract Analysis (=  Springer Undergraduate Mathematics Series . Volume 15 ). Springer Verlag, London ( inter alia ) 2002, ISBN 1-85233-424-X ( MR1870768 ).
  • MH Stone: Applications of the Theory of Boolean Rings to General Topology . In: Transactions of the American Mathematical Society , 41 (3), 1937, pp. 375-481, doi: 10.2307 / 1989788 .
  • MH Stone: The Generalized Weierstrass Approximation Theorem . In: Mathematics Magazine , 21 (4), 1948), pp. 167-184; 21 (5), pp. 237-254.
  • K. Weierstrass: About the analytical representability of so-called arbitrary functions of a real variable . In: Meeting reports of the Royal Prussian Academy of Sciences in Berlin , 1885 (II). ( First communication pp. 633–639, Second communication pp. 789–805.)

Web links

Individual evidence

  1. Elliot Ward Cheney: Introduction to Approximation Theory . McGraw-Hill Book Company, 1966, ISBN 0-07-010757-2 , p. 226
  2. Mícheál Ó Searcóid: Elements of Abstract Analysis. 2002, pp. 241-243