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| In [[mathematics]], the '''Paley–Wiener integral''' is a simple [[stochastic integral]]. When applied to [[Abstract Wiener space|classical Wiener space]], it is less general than the [[Itō integral]], but the two agree when they are both defined.
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| The integral is named after its discoverers, [[Raymond Paley]] and [[Norbert Wiener]].
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| ==Definition==
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| Let ''i'' : ''H'' → ''E'' be an [[abstract Wiener space]] with abstract Wiener measure ''γ'' on ''E''. Let ''j'' : ''E''<sup>∗</sup> → ''H'' be the [[adjoint of an operator|adjoint]] of ''i''. (We have abused notation slightly: strictly speaking, ''j'' : ''E''<sup>∗</sup> → ''H''<sup>∗</sup>, but since ''H'' is a [[Hilbert space]], it is [[Isometry|isometrically isomorphic]] to its [[dual space]] ''H''<sup>∗</sup>, by the [[Riesz representation theorem]].{{Citation needed|date=September 2010}})
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| It can be shown that ''j'' is an [[injective function]] and has [[dense (topology)|dense]] [[image (function)|image]] in ''H''.{{Citation needed|date=September 2010}} Furthermore, it can be shown that every [[linear functional]] ''f'' ∈ ''E''<sup>∗</sup> is also [[square-integrable]]: in fact,
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| :<math>\| f \|_{L^{2} (E, \gamma; \mathbb{R})} = \| j(f) \|_{H}</math>
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| This defines a natural [[linear map]] from ''j''(''E''<sup>∗</sup>) to ''L''<sup>2</sup>(''E'', ''γ''; '''R'''), under which ''j''(''f'') ∈ ''j''(''E''<sup>∗</sup>) ⊆ ''H'' goes to the [[equivalence class]] [''f''] of ''f'' in ''L''<sup>2</sup>(''E'', ''γ''; '''R'''). This is well-defined since ''j'' is injective. This map is an [[isometry]], so it is [[continuous function|continuous]].
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| However, since a continuous linear map between [[Banach space]]s such as ''H'' and ''L''<sup>2</sup>(''E'', ''γ''; '''R''') is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension ''I'' : ''H'' → ''L''<sup>2</sup>(''E'', ''γ''; '''R''') of the above natural map ''j''(''E''<sup>∗</sup>) → ''L''<sup>2</sup>(''E'', ''γ''; '''R''') to the whole of ''H''.
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| This isometry ''I'' : ''H'' → ''L''<sup>2</sup>(''E'', ''γ''; '''R''') is known as the '''Paley–Wiener map'''. ''I''(''h''), also denoted <''h'', −><sup>∼</sup>, is a function on ''E'' and is known as the '''Paley–Wiener integral''' (with respect to ''h'' ∈ ''H'').
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| It is important to note that the Paley–Wiener integral for a particular element ''h'' ∈ ''H'' is a [[Function (mathematics)|function]] on ''E''. The notation <''h'', ''x''><sup>∼</sup> does not really denote an inner product (since ''h'' and ''x'' belong to two different spaces), but is a convenient [[abuse of notation]] in view of the [[Cameron–Martin theorem]]. For this reason, many authors{{Citation needed|date=September 2010}} prefer to write <''h'', −><sup>∼</sup>(''x'') or ''I''(''h'')(''x'') rather than using the more compact but potentially confusing <''h'', ''x''><sup>∼</sup> notation.
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| ==See also==
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| Other stochastic integrals:
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| * [[Itō integral]]
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| * [[Skorokhod integral]]
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| * [[Stratonovich integral]]
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| {{No footnotes|date=September 2010}}
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| ==References==
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| *Park, C.; Skoug, D. (1988) "A Note on Paley-Wiener-Zygmund Stochastic Integrals", ''Proceedings of the American Mathematical Society', 103 (2), 591–601 {{JSTOR|2047184}}
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| *Elworthy, D. (2008) [http://www.tjsullivan.org.uk/pdf/MA482_Stochastic_Analysis.pdf ''MA482 Stochastic Analysis''], Lecture Notes, University of Warwick (Section 6)
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| {{DEFAULTSORT:Paley-Wiener Integral}}
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| [[Category:Definitions of mathematical integration]]
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| [[Category:Stochastic calculus]]
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