ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Hyperfunction

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about hyperfunctions in a mathematical context. For biological hypo- or hyperfunctions, see <a href="Endocrine_disease" title="Endocrine disease">endocrine disease</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, hyperfunctions are generalizations of functions, as a 'jump' from one <a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a> to another at a boundary, and can be thought of informally as <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> of infinite order. Hyperfunctions were introduced by <a href="Mikio_Sato" title="Mikio Sato">Mikio Sato</a> in <a href="#CITEREFSato1958">1958</a> in Japanese, (<a href="#CITEREFSato1959">1959</a>, <a href="#CITEREFSato1960">1960</a> in English), building upon earlier work by <a href="Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a>, <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck</a> and others.

<div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2></div>

A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the <a href="Upper_half-plane" title="Upper half-plane">upper half-plane</a> and another on the lower half-plane. That is, a hyperfunction is specified by a pair (f, g), where f is a holomorphic function on the upper half-plane and g is a holomorphic function on the lower half-plane.

Informally, the hyperfunction is what the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f-g}">

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<annotation encoding="application/x-tex">{\displaystyle f-g}</annotation>

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</math><img src="./e3f3019b383024c33e03b71a287d195f958ca89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f-g}" loading="lazy"> would be at the real line itself. This difference is not affected by adding the same holomorphic function to both f and g, so if h is a holomorphic function on the whole <a href="Complex_plane" title="Complex plane">complex plane</a>, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.

<div class="mw-heading mw-heading3"><h3 id="Definition_in_one_dimension">Definition in one dimension</h3></div>

The motivation can be concretely implemented using ideas from <a href="Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}">

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</math><img src="./d6ae2ed4058fb748a183d9ada8aea50a00d0c89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}" loading="lazy"> be the <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation>

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</math><img src="./8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" loading="lazy"> Define the hyperfunctions on the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> as the first <a href="Local_cohomology" title="Local cohomology">local cohomology</a> group:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(\mathbb {R} )=H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}}).}</annotation>

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</math><img src="./534a9bd483c93e16d6e21cd087bdc9a079c1c538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.497ex; height:3.176ex;" alt="{\displaystyle {\mathcal {B}}(\mathbb {R} )=H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}}).}" loading="lazy"></dd></dl>

Concretely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{+}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{+}}</annotation>

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</math><img src="./c79f3e776f6dfb76eb39c85ad01c12d06836b87a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{-}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{-}}</annotation>

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</math><img src="./ef850c756b367d9e303ae4fa5f58209cd63c3541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ^{-}}" loading="lazy"> be the <a href="Upper_half-plane" title="Upper half-plane">upper half-plane</a> and <a href="Lower_half-plane" class="mw-redirect" title="Lower half-plane">lower half-plane</a> respectively. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{+}\cup \mathbb {C} ^{-}=\mathbb {C} \setminus \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{+}\cup \mathbb {C} ^{-}=\mathbb {C} \setminus \mathbb {R} }</annotation>

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</math><img src="./2ede87b355e8f69dc4b25f070861a9923cfcf721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.61ex; height:3.009ex;" alt="{\displaystyle \mathbb {C} ^{+}\cup \mathbb {C} ^{-}=\mathbb {C} \setminus \mathbb {R} }" loading="lazy"> so

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<annotation encoding="application/x-tex">{\displaystyle H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}})=\left[H^{0}(\mathbb {C} ^{+},{\mathcal {O}})\oplus H^{0}(\mathbb {C} ^{-},{\mathcal {O}})\right]/H^{0}(\mathbb {C} ,{\mathcal {O}}).}</annotation>

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</math><img src="./973e93046c1d3bda18b6090c96598cde598a26a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.404ex; height:3.343ex;" alt="{\displaystyle H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}})=\left[H^{0}(\mathbb {C} ^{+},{\mathcal {O}})\oplus H^{0}(\mathbb {C} ^{-},{\mathcal {O}})\right]/H^{0}(\mathbb {C} ,{\mathcal {O}}).}" loading="lazy"></dd></dl>

Since the zeroth cohomology group of any sheaf is simply the global sections of that sheaf, we see that a hyperfunction is a pair of holomorphic functions one each on the upper and lower complex halfplane modulo entire holomorphic functions.

More generally one can define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(U)}</annotation>

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</math><img src="./0a8d66addc7549b88b3643b8f96bdf127c41ced2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.135ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(U)}" loading="lazy"> for any open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle U\subseteq \mathbb {R} }</annotation>

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</math><img src="./a3891fd33958c3b7dc0dbd3e6861db3fbefeb66e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.559ex; height:2.343ex;" alt="{\displaystyle U\subseteq \mathbb {R} }" loading="lazy"> as the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}({\tilde {U}}\setminus U,{\mathcal {O}})/H^{0}({\tilde {U}},{\mathcal {O}})}">

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<annotation encoding="application/x-tex">{\displaystyle H^{0}({\tilde {U}}\setminus U,{\mathcal {O}})/H^{0}({\tilde {U}},{\mathcal {O}})}</annotation>

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</math><img src="./3e0d0036aeff64d2db8ec462f00b4caa794cb8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.407ex; height:3.176ex;" alt="{\displaystyle H^{0}({\tilde {U}}\setminus U,{\mathcal {O}})/H^{0}({\tilde {U}},{\mathcal {O}})}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {U}}\subseteq \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {U}}\subseteq \mathbb {C} }</annotation>

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</math><img src="./1f6532d5c58617df2762f76d69679f88038020d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.559ex; height:2.843ex;" alt="{\displaystyle {\tilde {U}}\subseteq \mathbb {C} }" loading="lazy"> is any open set with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {U}}\cap \mathbb {R} =U}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {U}}\cap \mathbb {R} =U}</annotation>

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</math><img src="./9f741d729f0af2dd6d8c6c3a670e47a2e3f84027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.924ex; height:2.676ex;" alt="{\displaystyle {\tilde {U}}\cap \mathbb {R} =U}" loading="lazy">. One can show that this definition does not depend on the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {U}}}">

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</math><img src="./6d5ebcb074c33eaad0adcdb3beb42411ab56ba96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.676ex;" alt="{\displaystyle {\tilde {U}}}" loading="lazy"> giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>If f is any holomorphic function on the whole complex plane, then the restriction of f to the real axis is a hyperfunction, represented by either (f, 0) or (0, −f).</li>

<li>The <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> can be represented as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log(z),-{\frac {1}{2\pi i}}\log(z)\right).}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log(z),-{\frac {1}{2\pi i}}\log(z)\right).}</annotation>

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</math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(z)}">

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<annotation encoding="application/x-tex">{\displaystyle \log(z)}</annotation>

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</math><img src="./6febb7119dbf902cadafa39f3dcc3bd446ef1a50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.869ex; height:2.843ex;" alt="{\displaystyle \log(z)}" loading="lazy"> is the <a href="Complex_logarithm#Principal_value" title="Complex logarithm">principal value of the complex logarithm</a> of z.</li>

<li>The <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta "function"</a> is represented by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\dfrac {1}{2\pi iz}},{\dfrac {1}{2\pi iz}}\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \left({\dfrac {1}{2\pi iz}},{\dfrac {1}{2\pi iz}}\right).}</annotation>

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</math>This is really a restatement of <a href="Cauchy's_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a>. To verify it one can calculate the integration of f just below the real line, and subtract integration of g just above the real line - both from left to right. Note that the hyperfunction can be non-trivial, even if the components are analytic continuation of the same function. Also this can be easily checked by differentiating the Heaviside function.</li>

<li>If g is a <a href="Continuous_function" title="Continuous function">continuous function</a> (or more generally a <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a>) on the real line with support contained in a bounded interval I, then g corresponds to the hyperfunction (f, −f), where f is a holomorphic function on the complement of I defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.}</annotation>

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</math> This function f jumps in value by g(x) when crossing the real axis at the point x. The formula for f follows from the previous example by writing g as the <a href="Convolution" title="Convolution">convolution</a> of itself with the Dirac delta function.</li>

<li>Using a partition of unity one can write any continuous function (distribution) as a locally finite sum of functions (distributions) with compact support. This can be exploited to extend the above embedding to an embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}</annotation>

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</math><img src="./1205621aa96673193e91da7d815797a6df318207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.256ex; height:3.009ex;" alt="{\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}" loading="lazy"></li>

<li>If f is any function that is holomorphic everywhere except for an <a href="Essential_singularity" title="Essential singularity">essential singularity</a> at 0 (for example, e1/z), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,-f)}">

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</math><img src="./752bf064ce4af0f303cea39613748c90533b7163.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle (f,-f)}" loading="lazy"> is a hyperfunction with <a href="Support_(mathematics)" title="Support (mathematics)">support</a> 0 that is not a distribution. If f has a pole of finite order at 0 then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,-f)}">

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</math><img src="./752bf064ce4af0f303cea39613748c90533b7163.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle (f,-f)}" loading="lazy"> is a distribution, so when f has an essential singularity then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,-f)}">

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<annotation encoding="application/x-tex">{\displaystyle (f,-f)}</annotation>

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</math><img src="./752bf064ce4af0f303cea39613748c90533b7163.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle (f,-f)}" loading="lazy"> looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Operations_on_hyperfunctions">Operations on hyperfunctions</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq \mathbb {R} }">

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<ul><li>By definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(U)}">

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</math><img src="./0a8d66addc7549b88b3643b8f96bdf127c41ced2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.135ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(U)}" loading="lazy"> is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}">

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<annotation encoding="application/x-tex">{\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}</annotation>

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<li>The obvious restriction maps turn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation>

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</math><img src="./e5622de88a69f68340f8dcb43d0b8bd443ba9e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}" loading="lazy"> into a <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> (which is in fact <a href="Flabby_sheaf" class="mw-redirect" title="Flabby sheaf">flabby</a>).</li>

<li>Multiplication with real analytic functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in {\mathcal {O}}(U)}">

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</math><img src="./e182f731db9331023a965951ca7894a42da14ac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.621ex; height:2.843ex;" alt="{\displaystyle h\in {\mathcal {O}}(U)}" loading="lazy"> and differentiation are well-defined:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}}">

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</mtd>

<mtd>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

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</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}}</annotation>

</semantics>

</math> With these definitions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(U)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(U)}</annotation>

</semantics>

</math><img src="./0a8d66addc7549b88b3643b8f96bdf127c41ced2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.135ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(U)}" loading="lazy"> becomes a <a href="D-module" title="D-module">D-module</a> and the embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./b1fc3be913eee77947ac829889c8ffa5bae6daf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.927ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}" loading="lazy"> is a morphism of D-modules.</li>

<li>A point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in U}</annotation>

</semantics>

</math><img src="./cd1991ea9cb2ab076462a5538242321f0d0ee991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle a\in U}" loading="lazy"> is called a holomorphic point of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {B}}(U)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {B}}(U)}</annotation>

</semantics>

</math><img src="./263de97a7f1cfb8593c44466f0fb7c5ff4e72a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.254ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {B}}(U)}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> restricts to a real analytic function in some small neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a.}</annotation>

</semantics>

</math><img src="./6b803da9c45c1186883bde55107e9ccb102c92c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leqslant b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\leqslant b}</annotation>

</semantics>

</math><img src="./8f994f4b416430409bb184cfa927b85392ee9d18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leqslant b}" loading="lazy"> are two holomorphic points, then integration is well-defined: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz}">

</mrow>

</mrow>

</msubsup>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

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</msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz}</annotation>

</semantics>

</math> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }}</annotation>

</semantics>

</math><img src="./ca92246a71a816e569d9ea95e296d36ec391639d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.108ex; height:3.176ex;" alt="{\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }}" loading="lazy"> are arbitrary curves with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}</annotation>

</semantics>

</math><img src="./f1fad96a935efd04626e602b94405867c7df73bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.479ex; height:2.843ex;" alt="{\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}" loading="lazy"> The integrals are independent of the choice of these curves because the upper and lower half plane are <a href="Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>.</li>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{c}(U)}</annotation>

</semantics>

</math><img src="./70cb3d1deb10b528ac08a5be402795d3f4d3cdeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.063ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}_{c}(U)}" loading="lazy"> be the space of hyperfunctions with compact support. Via the bilinear form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\mathcal {B}}_{c}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}}">

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\mathcal {B}}_{c}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}}</annotation>

</semantics>

</math> one associates to each hyperfunction with compact support a continuous linear function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(U).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(U).}</annotation>

</semantics>

</math><img src="./aedb51a184b33a5f3c17b04d0f1c981636a2b2f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.089ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(U).}" loading="lazy"> This induces an identification of the dual space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}'(U),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}'(U),}</annotation>

</semantics>

</math><img src="./e4c6db4cc33f1c0cd53c462121189f2bad113f57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.773ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}'(U),}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}_{c}(U).}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}_{c}(U).}</annotation>

</semantics>

</math><img src="./d7892e2120f585da91f346863613cccf8f7418bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}_{c}(U).}" loading="lazy"> A special case worth considering is the case of continuous functions or distributions with compact support: If one considers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{c}^{0}(U)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{c}^{0}(U)}</annotation>

</semantics>

</math><img src="./70f77e4c638ed29f0653cec46782f5dbda6a4d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:3.009ex;" alt="{\displaystyle C_{c}^{0}(U)}" loading="lazy"> (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}'(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}'(U)}</annotation>

</semantics>

</math><img src="./fcd4526cfb7f21e729970f4b540f5d6e52697bb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.613ex; height:3.009ex;" alt="{\displaystyle {\mathcal {E}}'(U)}" loading="lazy">) as a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(U)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(U)}</annotation>

</semantics>

</math><img src="./0a8d66addc7549b88b3643b8f96bdf127c41ced2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.135ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(U)}" loading="lazy"> via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathcal {E}}'(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in {\mathcal {E}}'(U)}</annotation>

</semantics>

</math><img src="./206b86ce3f4abc8d2176ddea0b3b8eb50ef693de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.783ex; height:3.009ex;" alt="{\displaystyle u\in {\mathcal {E}}'(U)}" loading="lazy"> is a distribution with compact support, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in {\mathcal {O}}(U)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \in {\mathcal {O}}(U)}</annotation>

</semantics>

</math><img src="./5c1dda4d4ce84b499d9da7653e831de8560d5e64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.803ex; height:2.843ex;" alt="{\displaystyle \varphi \in {\mathcal {O}}(U)}" loading="lazy"> is a real analytic function, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {supp} (u)\subset (a,b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {supp} (u)\subset (a,b)}</annotation>

</semantics>

</math><img src="./ba6630a7f1337d1a939693c87d0af7eee21c6fee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.102ex; height:2.843ex;" alt="{\displaystyle \operatorname {supp} (u)\subset (a,b)}" loading="lazy"> then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}</annotation>

</semantics>

</math> Thus this notion of integration gives a precise meaning to formal expressions like <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}\delta (x)\,dx}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}\delta (x)\,dx}</annotation>

</semantics>

</math> which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}</annotation>

</semantics>

</math><img src="./e1f5591ee330d330ad93f75e706cf1cacffeaaf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.55ex; height:3.009ex;" alt="{\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}" loading="lazy"> is a subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}'(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}'(U)}</annotation>

</semantics>

</math><img src="./34cc5a5e7b9ba63ce4b938b4c08f479c50b4f95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.127ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}'(U)}" loading="lazy">. This is an alternative description of the same embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}</annotation>

</semantics>

</math><img src="./3f7127baba2b7f41589997dd57460479f8944c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.471ex; height:2.509ex;" alt="{\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}" loading="lazy">.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi :U\to V}</annotation>

</semantics>

</math><img src="./989740e3470972d659d72a630c41998e784d91ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.799ex; height:2.176ex;" alt="{\displaystyle \Phi :U\to V}" loading="lazy"> is a real analytic map between open sets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">, then composition with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"> is a well-defined operator from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(V)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(V)}</annotation>

</semantics>

</math><img src="./8ff6a1e24815720efc74570a957b57d198a7e504.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.14ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(V)}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(U)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(U)}</annotation>

</semantics>

</math><img src="./0a8d66addc7549b88b3643b8f96bdf127c41ced2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.135ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(U)}" loading="lazy">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \Phi :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ \Phi :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}</annotation>

</semantics>

</math></li></ul>

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<ul><li><a href="Algebraic_analysis" title="Algebraic analysis">Algebraic analysis</a></li>

<li><a href="Generalized_function" title="Generalized function">Generalized function</a></li>

<li><a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution (mathematics)</a></li>

<li><a href="Microlocal_analysis" title="Microlocal analysis">Microlocal analysis</a></li>

<li><a href="Pseudo-differential_operator" title="Pseudo-differential operator">Pseudo-differential operator</a></li>

<li><a href="Sheaf_cohomology" title="Sheaf cohomology">Sheaf cohomology</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFImai2012" class="citation cs2"><a href="Isao_Imai_(physicist)" title="Isao Imai (physicist)">Imai, Isao</a> (2012) [1992], <a rel="nofollow" class="external text" href="https://www.springer.com/book/9780792315070">Applied Hyperfunction Theory</a>, Mathematics and its Applications (Book 8), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-010-5125-5</bdi></cite>.</li>

<li><cite id="CITEREFKaneko1988" class="citation cs2">Kaneko, Akira (1988), <a rel="nofollow" class="external text" href="https://www.springer.com/book/9789027728371">Introduction to the Theory of Hyperfunctions</a>, Mathematics and its Applications (Japanese Series, Vol. 3), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-277-2837-1</bdi></cite></li>

<li><cite id="CITEREFKashiwaraKawaiKimura2017" class="citation cs2 cs1-prop-long-vol"><a href="Masaki_Kashiwara" title="Masaki Kashiwara">Kashiwara, Masaki</a>; Kawai, Takahiro; Kimura, Tatsuo (2017) [1986], <a rel="nofollow" class="external text" href="https://press.princeton.edu/titles/798.html">Foundations of Algebraic Analysis</a>, Princeton Legacy Library (Book 5158), vol. PMS-37, translated by Kato, Goro (Reprint ed.), Princeton University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-62832-5</bdi></cite></li>

<li><cite id="CITEREFKomatsu1973" class="citation cs2">Komatsu, Hikosaburo, ed. (1973), <a rel="nofollow" class="external text" href="https://www.springer.com/book/9783540062189">Hyperfunctions and Pseudo-Differential Equations, Proceedings of a Conference at Katata, 1971</a>, Lecture Notes in Mathematics 287, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-06218-9</bdi></cite>.

<ul><li><cite id="CITEREFKomatsu" class="citation cs2">Komatsu, Hikosaburo, Relative cohomology of sheaves of solutions of differential equations, pp. 192–261</cite>.</li>

<li><cite id="CITEREFSatoKawaiKashiwara" class="citation cs2">Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki, Microfunctions and pseudo-differential equations, pp. 265–529</cite>. - It is called SKK.</li></ul></li>

<li><cite id="CITEREFMartineau1960–1961" class="citation cs2"><a href="Andr%C3%A9_Martineau" title="André Martineau">Martineau, André</a> (1960–1961), <a rel="nofollow" class="external text" href="http://www.numdam.org/item/SB_1960-1961__6__127_0">Les hyperfonctions de M. Sato</a>, Séminaire Bourbaki, Tome 6 (1960-1961), Exposé no. 214, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1611794">1611794</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0122.34902">0122.34902</a></cite>.</li>

<li><cite id="CITEREFMorimoto1993" class="citation cs2">Morimoto, Mitsuo (1993), <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontosa0000mori">An Introduction to Sato's Hyperfunctions</a>, Translations of Mathematical Monographs (Book 129), American Mathematical Society, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-82184571-4</bdi></cite>.</li>

<li><cite id="CITEREFPham1975" class="citation cs2">Pham, F. L., ed. (1975), <a rel="nofollow" class="external text" href="https://www.springer.com/book/9783540071518">Hyperfunctions and Theoretical Physics, Rencontre de Nice, 21-30 Mai 1973</a>, Lecture Notes in Mathematics 449, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-37454-1</bdi></cite>.

<ul><li><cite id="CITEREFCerezoPiriouChazarain" class="citation cs2">Cerezo, A.; Piriou, A.; Chazarain, J., Introduction aux hyperfonctions, pp. 1–53</cite>.</li></ul></li>

<li><cite id="CITEREFSato1958" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Mikio_Sato" title="Mikio Sato">Sato, Mikio</a> (1958), <a rel="nofollow" class="external text" href="https://www.jstage.jst.go.jp/article/sugaku1947/10/1/10_1_1/_article/-char/ja/">"Cyōkansū no riron (Theory of Hyperfunctions)"</a>, Sūgaku (in Japanese), 10 (1), Mathematical Society of Japan: 1–27, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.11429%2Fsugaku1947.10.1">10.11429/sugaku1947.10.1</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0039-470X">0039-470X</a></cite></li>

<li><cite id="CITEREFSato1959" class="citation cs2">Sato, Mikio (1959), "Theory of Hyperfunctions, I", Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 8 (1): 139–193, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2261%2F6027">2261/6027</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0114124">0114124</a></cite>.</li>

<li><cite id="CITEREFSato1960" class="citation cs2">Sato, Mikio (1960), "Theory of Hyperfunctions, II", Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 8 (2): 387–437, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2261%2F6031">2261/6031</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0132392">0132392</a></cite>.</li>

<li><cite id="CITEREFSchapira1970" class="citation cs2"><a href="Pierre_Schapira_(mathematician)" title="Pierre Schapira (mathematician)">Schapira, Pierre</a> (1970), <a rel="nofollow" class="external text" href="https://www.springer.com/book/9783540049159">Theories des Hyperfonctions</a>, Lecture Notes in Mathematics 126, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-04915-9</bdi></cite>.</li>

<li><cite id="CITEREFSchlichtkrull2013" class="citation cs2">Schlichtkrull, Henrik (2013) [1984], <a rel="nofollow" class="external text" href="https://www.springer.com/book/9781461297758">Hyperfunctions and Harmonic Analysis on Symmetric Spaces</a>, Progress in Mathematics (Softcover reprint of the original 1st ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-9775-8</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1">Jacobs, Bryan. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Hyperfunction.html">"Hyperfunction"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite id="CITEREFKaneko2001" class="citation cs2">Kaneko, A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Hyperfunction">"Hyperfunction"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li></ul>

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