Wiener theorem

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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

Individual evidence

  1. Norbert Wiener: Tauberian theorems . In: Ann. of Math. , 33 (2), pp. 1-100
  2. ^ Sterling K. Berberian: Lectures in Functional Analysis and Operator Theory. 1974, p. 1 ff, p. 267 ff
  3. a b M. A. Neumark: Standardized Algebras. 1990, p. 221
  4. a b Kōsaku Yosida: Functional Analysis. 1980, p. 301
  5. Berberian, op.cit., P. 1
  6. Berberian, op. Cit., Pp. 1-10
  7. ^ DJ Newman: A simple proof of Wiener's 1 / f theorem . In: Proc. Amer. Math. Soc. , 48, pp. 264-265
  8. Berberian, op. Cit., Pp. 267-269
  9. Russian Математический сборник