[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/paley-wiener-integral-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki8\/paley-wiener-integral-wikipedia\/","headline":"Paley\u2013Wiener integral – Wikipedia","name":"Paley\u2013Wiener integral – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In mathematics, the Paley\u2013Wiener integral is a simple stochastic integral. When applied to","datePublished":"2016-06-11","dateModified":"2016-06-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki8\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/703e039b50db91d2980e1d722b0071d59b7f72e6","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/703e039b50db91d2980e1d722b0071d59b7f72e6","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki8\/paley-wiener-integral-wikipedia\/","about":["Wiki"],"wordCount":4930,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, the Paley\u2013Wiener integral is a simple stochastic integral. When applied to classical Wiener space, it is less general than the It\u014d integral, but the two agree when they are both defined.The integral is named after its discoverers, Raymond Paley and Norbert Wiener.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Definition[edit]Let i:H\u2192E{displaystyle i:Hto E} be an abstract Wiener space with abstract Wiener measure \u03b3{displaystyle gamma } on E{displaystyle E}. Let        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4j:E\u2217\u2192H{displaystyle j:E^{*}to H} be the adjoint of i{displaystyle i}. (We have abused notation slightly: strictly speaking, j:E\u2217\u2192H\u2217{displaystyle j:E^{*}to H^{*}}, but since H{displaystyle H} is a Hilbert space, it is isometrically isomorphic to its dual space H\u2217{displaystyle H^{*}}, by the Riesz representation theorem.)It can be shown that j{displaystyle j} is an injective function and has dense image in H{displaystyle H}.[citation needed] Furthermore, it can be shown that every linear functional f\u2208E\u2217{displaystyle fin E^{*}} is also square-integrable: in fact,\u2016f\u2016L2(E,\u03b3;R)=\u2016j(f)\u2016H{displaystyle |f|_{L^{2}(E,gamma ;mathbb {R} )}=|j(f)|_{H}}This defines a natural linear map from j(E\u2217){displaystyle j(E^{*})} to L2(E,\u03b3;R){displaystyle L^{2}(E,gamma ;mathbb {R} )}, under which j(f)\u2208j(E\u2217)\u2286H{displaystyle j(f)in j(E^{*})subseteq H} goes to the equivalence class [f]{displaystyle [f]} of f{displaystyle f} in L2(E,\u03b3;R){displaystyle L^{2}(E,gamma ;mathbb {R} )}. This is well-defined since j{displaystyle j} is injective. This map is an isometry, so it is continuous.However, since a continuous linear map between Banach spaces such as H{displaystyle H} and L2(E,\u03b3;R){displaystyle L^{2}(E,gamma ;mathbb {R} )} is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension I:H\u2192L2(E,\u03b3;R){displaystyle I:Hto L^{2}(E,gamma ;mathbb {R} )} of the above natural map j(E\u2217)\u2192L2(E,\u03b3;R){displaystyle j(E^{*})to L^{2}(E,gamma ;mathbb {R} )} to the whole of H{displaystyle H}.This isometry I:H\u2192L2(E,\u03b3;R){displaystyle I:Hto L^{2}(E,gamma ;mathbb {R} )} is known as the Paley\u2013Wiener map. I(h){displaystyle I(h)}, also denoted \u27e8h,x\u27e9\u223c{displaystyle langle h,xrangle ^{sim }}, is a function on E{displaystyle E} and is known as the Paley\u2013Wiener integral (with respect to h\u2208H{displaystyle hin H}).It is important to note that the Paley\u2013Wiener integral for a particular element h\u2208H{displaystyle hin H} is a function on E{displaystyle E}. The notation \u27e8h,x\u27e9\u223c{displaystyle langle h,xrangle ^{sim }} does not really denote an inner product (since h{displaystyle h} and x{displaystyle x} belong to two different spaces), but is a convenient abuse of notation in view of the Cameron\u2013Martin theorem. For this reason, many authors[citation needed] prefer to write \u27e8h,\u2212\u27e9\u223c(x){displaystyle langle h,-rangle ^{sim }(x)} or I(h)(x){displaystyle I(h)(x)} rather than using the more compact but potentially confusing \u27e8h,x\u27e9\u223c{displaystyle langle h,xrangle ^{sim }} notation.See also[edit]Other stochastic integrals:References[edit]Park, Chull; Skoug, David (1988), “A Note on Paley-Wiener-Zygmund Stochastic Integrals”, Proceedings of the American Mathematical Society, 103 (2): 591\u2013601, doi:10.1090\/S0002-9939-1988-0943089-8, JSTOR\u00a02047184Elworthy, David (2008), MA482 Stochastic Analysis (PDF), Lecture Notes, University of Warwick (Section 6)       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/paley-wiener-integral-wikipedia\/#breadcrumbitem","name":"Paley\u2013Wiener integral – Wikipedia"}}]}]