Wiener theorem
The set of Wiener ( English Wiener's theorem or Wiener's theorem ) is a classic mathematical theorem , the field in the transition between the areas of the harmonic analysis and the functional analysis is based. It goes back to a work by the American mathematician Norbert Wiener from 1932 and deals with the question of the series expandability of reciprocal values of certain Fourier series .
Formulation of the sentence
According to the presentation of the American mathematician Sterling K. Berberian , Wiener’s theorem can be formulated as follows:
- The reciprocal of a non-vanishing , absolutely convergent trigonometric series is always itself an absolutely convergent trigonometric series.
- In other words it applies:
-
Is a sequence of complex numbers with
-
and owns the through
-
If the defined complex-valued function does not have a zero , there is a sequence of complex numbers such that
-
applies and at the same time the function in the form resulting from the formation of the reciprocal value
- is representable.
For background and evidence
In his textbook Lectures in Functional Analysis and Operator Theory, Sterling K. Berberian follows the proof of IM Gel'fand from 1941 and emphasizes in this context that this proof Gel'fand an early triumph of the functional analytical approach (“early triumph of the functional-analytic point of view "). There are also numerous other proofs, including an elementary proof by Donald Joseph Newman (1930–2007). Wiener's theorem also results as a corollary from more extensive theorems of the theory of commutative Banach algebras .
literature
- Sterling K. Berberian : Lectures in Functional Analysis and Operator Theory (= Graduate Texts in Mathematics . Volume 15 ). Springer Verlag, New York / Heidelberg / Berlin 1974, ISBN 0-387-90080-2 ( MR0417727 ).
- IM Gel'fand: About absolutely convergent trigonometric series and integrals . In: Matematitscheskii sbornik (NS) . tape 9 (51) , 1941, pp. 51-66 ( MR0004727 ).
- MA Neumark : Standardized Algebras . Verlag Harri Deutsch, Thun / Frankfurt / Main 1990, ISBN 3-8171-1001-4 ( MR1038909 ).
- DJ Newman: A simple proof of Wiener's 1 / f theorem . In: Proceedings of the American Mathematical Society . tape 48 , 1975, pp. 264-265 ( MR0365002 ).
- Norbert Wiener: Tauberian Theorems . In: Annals of Mathematics . tape 33 (2) , 1932, pp. 1-100 ( MR1503035 ).
- Kōsaku Yosida : Functional Analysis (= Basic Teachings of the Mathematical Sciences . Volume 123 ). 6th edition. Springer Verlag, New York / Heidelberg / Berlin 1980, ISBN 3-540-10210-8 .
Individual evidence
- ↑ Norbert Wiener: Tauberian theorems . In: Ann. of Math. , 33 (2), pp. 1-100
- ^ Sterling K. Berberian: Lectures in Functional Analysis and Operator Theory. 1974, p. 1 ff, p. 267 ff
- ↑ a b M. A. Neumark: Standardized Algebras. 1990, p. 221
- ↑ a b Kōsaku Yosida: Functional Analysis. 1980, p. 301
- ↑ Berberian, op.cit., P. 1
- ↑ Berberian, op. Cit., Pp. 1-10
- ^ DJ Newman: A simple proof of Wiener's 1 / f theorem . In: Proc. Amer. Math. Soc. , 48, pp. 264-265
- ↑ Berberian, op. Cit., Pp. 267-269
- ↑ Russian Математический сборник