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| In [[mathematical analysis]], '''Wiener's tauberian theorem''' is any of several related results proved by [[Norbert Wiener]] in 1932.<ref>See {{harvtxt|Wiener|1932}}.</ref> They provide a necessary and sufficient condition under which any function in [[Lp space|{{math|''L''<sub>1</sub>}} or {{math|''L''<sub>2</sub>}}]] can be approximated by [[linear combination]]s of [[shift operator|translations]] of a given function.<ref>see {{harvtxt|Rudin|1991}}.</ref>
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| Informally, if the [[Fourier transform]] of a function {{math|''f''}} vanishes on a certain set {{math|''Z''}}, the Fourier transform of any linear combination of translations of {{math|''f''}} also vanishes on {{math|''Z''}}. Therefore the linear combinations of translations of {{math|''f''}} can not approximate a function whose Fourier transform does not vanish on {{math|''Z''}}.
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| Wiener's theorems make this precise, stating that linear combinations of translations of {{math|''f''}} are [[dense set|dense]] if and only the [[zero set]] of the Fourier transform of {{math|''f''}} is empty (in the case of {{math|''L''<sub>1</sub>}}) or of Lebesgue measure zero (in the case of {{math|''L''<sub>2</sub>}}).
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| ==The condition in {{math|''L''<sub>1</sub>}}==
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| Let {{math|''f'' ∈ ''L''<sub>1</sub>('''R''')}} be an integrable function. The [[linear span|span]] of translations {{math|''f''<sub>''a''</sub>(''x'')}} = {{math|''f''(''x'' + ''a'')}} is dense in {{math|''L''<sub>1</sub>('''R''')}} if and only if the Fourier transform of {{math|''f''}} has no real zeros.
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| ===Tauberian reformulation===
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| The following statement is equivalent to the previous result, and explains why Wiener's result is a [[Tauberian theorem]]:
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| Suppose the Fourier transform of {{math|''f'' ∈ ''L''<sub>1</sub>}} has no real zeros, and suppose the convolution {{math|''f'' * ''h''}} tends to zero at infinity for some {{math|''h'' ∈ ''L''<sub>∞</sub>}}. Then the convolution {{math|''g'' * ''h''}} tends to zero at infinity for any {{math|''g'' ∈ ''L''<sub>1</sub>}}.
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| More generally, if
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| : <math> \lim_{x \to \infty} (f*h)(x) = A \int f(x) \, dx </math>
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| for some {{math|''f'' ∈ ''L''<sub>1</sub>}} the Fourier transform of which has no real zeros, then also
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| : <math> \lim_{x \to \infty} (g*h)(x) = A \int g(x) \, dx </math>
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| for any {{math|''g'' ∈ ''L''<sub>1</sub>}}.
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| ===Discrete version===
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| Wiener's theorem has a counterpart in {{math|''l''<sub>1</sub>('''Z''')}}: the span of the translations of {{math|''f'' ∈ ''l''<sub>1</sub>('''Z''')}} is dense if and only if the Fourier transform
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| :<math> \varphi(\theta) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta} \, </math>
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| has no real zeros. The following statements are equivalent version of this result:
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| * Suppose the Fourier transform of {{math|''f'' ∈ ''l''<sub>1</sub>('''Z''')}} has no real zeros, and the convolution {{math|''f'' * ''h''}} tends to zero at infinity for some bounded sequence {{math|''h''}}. Then {{math|''g'' * ''h''}} for any {{math|''g'' ∈ ''l''<sub>1</sub>('''Z''')}}.
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| * Let {{math|''φ''}} be a function on the unit circle with absolutely convergent Fourier series. Then {{math|1/''φ''}} has absolutely convergent Fourier series if and only if {{math|''φ''}} has no zeros.
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| {{harvs|txt|last=Gelfand|author-link=Israel Gelfand|year1=1941a|year2=1941b}} showed that this is equivalent to the following property of the [[Wiener algebra]] {{math|''A''('''T''')}}, which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:
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| * The maximal ideals of {{math|''A''('''T''')}} are all of the form
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| ::<math> M_x = \left\{ f \in A(\mathbb{T}) \, \mid \, f(x) = 0 \right\}, \quad x \in \mathbb{T}. \,</math> | |
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| ==The condition in {{math|''L''<sub>2</sub>}}==
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| Let {{math|''f'' ∈ ''L''<sub>2</sub>('''R''')}} be a square-integrable function. The span of translations {{math|''f''<sub>''a''</sub>(''x'')}} = {{math|''f''(''x'' + ''a'')}} is dense in {{math|''L''<sub>2</sub>('''R''')}} if and only if the real zeros of the Fourier transform of {{math|''f''}} form a set of zero [[Lebesgue measure]].
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| The parallel statement in {{math|''l''<sub>2</sub>('''Z''')}} is as follows: the span of translations of a sequence {{math|''f'' ∈ ''l''<sub>2</sub>('''Z''')}} is dense if and only if the zero set of the Fourier transform
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| :<math> \varphi(\theta) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta} \, </math>
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| has zero Lebesgue measure.
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{Citation | last1=Gelfand | first1=I. | author-link=Israel Gelfand|title=Normierte Ringe | year=1941a | journal=Rec. Math. (Mat. Sbornik) N.S.| volume=9 (51) | pages=3–24 | mr=0004726}}
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| *{{Citation | last1=Gelfand | first1=I. | author-link=Israel Gelfand|title=Über absolut konvergente trigonometrische Reihen und Integrale | year=1941b | journal=Rec. Math. (Mat. Sbornik) N.S.| volume=9 (51) | pages=51–66 | mr=0004727}}
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| *{{Citation | mr=1157815 | last=Rudin | first = W.| author-link=Walter Rudin|title=Functional analysis|series=International Series in Pure and Applied Mathematics|publisher=McGraw-Hill, Inc.|location=New York|year=1991|isbn=0-07-054236-8}}
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| *{{Citation | last=Wiener|first=N.|author-link=Norbert Wiener|title=Tauberian Theorems|journal=Annals of Math.|volume=33|issue=1|year=1932|pages=1–100|jstor=1968102}}
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| ==External links==
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| *{{eom|id=W/w097950|title=Wiener Tauberian theorem|first=A.I.|last=Shtern}}
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| [[Category:Real analysis]]
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| [[Category:Harmonic analysis]]
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| [[Category:Theorems in analysis]]
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| [[Category:Tauberian theorems]]
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Hi there, I am Alyson Pomerleau and I believe it sounds quite great when you say it. Alaska is exactly where he's usually been residing. It's not a typical thing but what I like doing is to climb but I don't have the time lately. I am presently a journey agent.
Also visit my homepage; online reader