Hyperfunction

<h2 id="formulation">Formulation</h2>
A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the <a href="/facts/Upper_half-plane/7fPqrowr">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 
 
 
 
 f
 −
 g
 
 
 {\displaystyle f-g}
 
 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="/facts/Complex_plane/vE0QrxJp">complex plane</a>, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.

<h3>Definition in one dimension</h3>
The motivation can be concretely implemented using ideas from <a href="/facts/Sheaf_cohomology/TrcpmD6k">sheaf cohomology</a>. Let 
 
 
 
 
 
 O
 
 
 
 
 {\displaystyle {\mathcal {O}}}
 
 be the <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> of <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> on 
 
 
 
 
 C
 
 .
 
 
 {\displaystyle \mathbb {C} .}
 
 Define the hyperfunctions on the <a href="/facts/Real_line/6eq42xX5">real line</a> as the first <a href="/facts/Local_cohomology/NjY1n2Vl">local cohomology</a> group:

B
          
        
        (
        
          R
        
        )
        =
        
          H
          
            
              R
            
          
          
            1
          
        
        (
        
          C
        
        ,
        
          
            O
          
        
        )
        .
      
    
    {\displaystyle {\mathcal {B}}(\mathbb {R} )=H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}}).}

Concretely, let 
 
 
 
 
 
 C
 
 
 +
 
 
 
 
 {\displaystyle \mathbb {C} ^{+}}
 
 and 
 
 
 
 
 
 C
 
 
 −
 
 
 
 
 {\displaystyle \mathbb {C} ^{-}}
 
 be the <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a> and <a href="/facts/Lower_half-plane/7fPqrowr">lower half-plane</a> respectively. Then 
 
 
 
 
 
 C
 
 
 +
 
 
 ∪
 
 
 C
 
 
 −
 
 
 =
 
 C
 
 ∖
 
 R
 
 
 
 {\displaystyle \mathbb {C} ^{+}\cup \mathbb {C} ^{-}=\mathbb {C} \setminus \mathbb {R} }
 
 so

H
          
            
              R
            
          
          
            1
          
        
        (
        
          C
        
        ,
        
          
            O
          
        
        )
        =
        
          [
          
            
              H
              
                0
              
            
            (
            
              
                C
              
              
                +
              
            
            ,
            
              
                O
              
            
            )
            ⊕
            
              H
              
                0
              
            
            (
            
              
                C
              
              
                −
              
            
            ,
            
              
                O
              
            
            )
          
          ]
        
        
          /
        
        
          H
          
            0
          
        
        (
        
          C
        
        ,
        
          
            O
          
        
        )
        .
      
    
    {\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}}).}

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 
 
 
 
 
 
 B
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {B}}(U)}
 
 for any open set 
 
 
 
 U
 ⊆
 
 R
 
 
 
 {\displaystyle U\subseteq \mathbb {R} }
 
 as the quotient 
 
 
 
 
 H
 
 0
 
 
 (
 
 
 
 U
 ~
 
 
 
 ∖
 U
 ,
 
 
 O
 
 
 )
 
 /
 
 
 H
 
 0
 
 
 (
 
 
 
 U
 ~
 
 
 
 ,
 
 
 O
 
 
 )
 
 
 {\displaystyle H^{0}({\tilde {U}}\setminus U,{\mathcal {O}})/H^{0}({\tilde {U}},{\mathcal {O}})}
 
 where 
 
 
 
 
 
 
 U
 ~
 
 
 
 ⊆
 
 C
 
 
 
 {\displaystyle {\tilde {U}}\subseteq \mathbb {C} }
 
 is any open set with 
 
 
 
 
 
 
 U
 ~
 
 
 
 ∩
 
 R
 
 =
 U
 
 
 {\displaystyle {\tilde {U}}\cap \mathbb {R} =U}
 
. One can show that this definition does not depend on the choice of 
 
 
 
 
 
 
 U
 ~
 
 
 
 
 
 {\displaystyle {\tilde {U}}}
 
 giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.

<h2 id="examples">Examples</h2>
<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="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a> can be represented as 
 
 
 
 H
 (
 x
 )
 =
 
 (
 
 1
 −
 
 
 1
 
 2
 π
 i
 
 
 
 log
 ⁡
 (
 z
 )
 ,
 −
 
 
 1
 
 2
 π
 i
 
 
 
 log
 ⁡
 (
 z
 )
 
 )
 
 .
 
 
 {\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log(z),-{\frac {1}{2\pi i}}\log(z)\right).}
 
 where 
 
 
 
 log
 ⁡
 (
 z
 )
 
 
 {\displaystyle \log(z)}
 
 is the <a href="/facts/Complex_logarithm/X2U0YQuG">principal value of the complex logarithm</a> of z.</li>
<li>The <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta "function"</a> is represented by 
 
 
 
 
 (
 
 
 
 
 1
 
 2
 π
 i
 z
 
 
 
 
 ,
 
 
 
 1
 
 2
 π
 i
 z
 
 
 
 
 
 )
 
 .
 
 
 {\displaystyle \left({\dfrac {1}{2\pi iz}},{\dfrac {1}{2\pi iz}}\right).}
 
This is really a restatement of <a href="/facts/Cauchy%2527s_integral_formula/4B35nkvV">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="/facts/Continuous_function/sKbl02pB">continuous function</a> (or more generally a <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">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 
 
 
 
 f
 (
 z
 )
 =
 
 
 1
 
 2
 π
 i
 
 
 
 
 ∫
 
 x
 ∈
 I
 
 
 g
 (
 x
 )
 
 
 1
 
 z
 −
 x
 
 
 
 
 d
 x
 .
 
 
 {\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.}
 
 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="/facts/Convolution/PVPrdz9J">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 
 
 
 
 
 
 
 
 D
 
 
 ′
 
 (
 
 R
 
 )
 →
 
 
 B
 
 
 (
 
 R
 
 )
 .
 
 
 
 {\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}
 
</li>
<li>If f is any function that is holomorphic everywhere except for an <a href="/facts/Essential_singularity/KMb1R292">essential singularity</a> at 0 (for example, e1/z), then 
 
 
 
 (
 f
 ,
 −
 f
 )
 
 
 {\displaystyle (f,-f)}
 
 is a hyperfunction with <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> 0 that is not a distribution. If f has a pole of finite order at 0 then 
 
 
 
 (
 f
 ,
 −
 f
 )
 
 
 {\displaystyle (f,-f)}
 
 is a distribution, so when f has an essential singularity then 
 
 
 
 (
 f
 ,
 −
 f
 )
 
 
 {\displaystyle (f,-f)}
 
 looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)</li></ul>
<h2 id="operations-on-hyperfunctions">Operations on hyperfunctions</h2>
Let 
 
 
 
 U
 ⊆
 
 R
 
 
 
 {\displaystyle U\subseteq \mathbb {R} }
 
 be any open subset.

<ul><li>By definition 
 
 
 
 
 
 B
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {B}}(U)}
 
 is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly: 
 
 
 
 a
 (
 
 f
 
 +
 
 
 ,
 
 f
 
 −
 
 
 )
 +
 b
 (
 
 g
 
 +
 
 
 ,
 
 g
 
 −
 
 
 )
 :=
 (
 a
 
 f
 
 +
 
 
 +
 b
 
 g
 
 +
 
 
 ,
 a
 
 f
 
 −
 
 
 +
 b
 
 g
 
 −
 
 
 )
 
 
 {\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}
 
</li>
<li>The obvious restriction maps turn 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 into a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> (which is in fact <a href="/facts/Flabby_sheaf/ykaN5Ivq">flabby</a>).</li>
<li>Multiplication with real analytic functions 
 
 
 
 h
 ∈
 
 
 O
 
 
 (
 U
 )
 
 
 {\displaystyle h\in {\mathcal {O}}(U)}
 
 and differentiation are well-defined:
 
 
 
 
 
 
 
 h
 (
 
 f
 
 +
 
 
 ,
 
 f
 
 −
 
 
 )
 
 
 
 :=
 (
 h
 
 f
 
 +
 
 
 ,
 h
 
 f
 
 −
 
 
 )
 
 
 
 
 
 
 d
 
 d
 z
 
 
 
 (
 
 f
 
 +
 
 
 ,
 
 f
 
 −
 
 
 )
 
 
 
 :=
 
 (
 
 
 
 
 d
 
 f
 
 +
 
 
 
 
 d
 z
 
 
 
 ,
 
 
 
 d
 
 f
 
 −
 
 
 
 
 d
 z
 
 
 
 
 )
 
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}}
 
 With these definitions 
 
 
 
 
 
 B
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {B}}(U)}
 
 becomes a <a href="/facts/D-module/l1y6evYe">D-module</a> and the embedding 
 
 
 
 
 
 
 D
 
 
 ′
 
 ↪
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}
 
 is a morphism of D-modules.</li>
<li>A point 
 
 
 
 a
 ∈
 U
 
 
 {\displaystyle a\in U}
 
 is called a holomorphic point of 
 
 
 
 f
 ∈
 
 
 B
 
 
 (
 U
 )
 
 
 {\displaystyle f\in {\mathcal {B}}(U)}
 
 if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 restricts to a real analytic function in some small neighbourhood of 
 
 
 
 a
 .
 
 
 {\displaystyle a.}
 
 If 
 
 
 
 a
 ⩽
 b
 
 
 {\displaystyle a\leqslant b}
 
 are two holomorphic points, then integration is well-defined: 
 
 
 
 
 ∫
 
 a
 
 
 b
 
 
 f
 :=
 −
 
 ∫
 
 
 γ
 
 +
 
 
 
 
 
 f
 
 +
 
 
 (
 z
 )
 
 d
 z
 +
 
 ∫
 
 
 γ
 
 −
 
 
 
 
 
 f
 
 −
 
 
 (
 z
 )
 
 d
 z
 
 
 {\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz}
 
 where 
 
 
 
 
 γ
 
 ±
 
 
 :
 [
 0
 ,
 1
 ]
 →
 
 
 C
 
 
 ±
 
 
 
 
 {\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }}
 
 are arbitrary curves with 
 
 
 
 
 γ
 
 ±
 
 
 (
 0
 )
 =
 a
 ,
 
 γ
 
 ±
 
 
 (
 1
 )
 =
 b
 .
 
 
 {\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}
 
 The integrals are independent of the choice of these curves because the upper and lower half plane are <a href="/facts/Simply_connected/xFlta63J">simply connected</a>.</li>
<li>Let 
 
 
 
 
 
 
 B
 
 
 
 c
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {B}}_{c}(U)}
 
 be the space of hyperfunctions with compact support. Via the bilinear form 
 
 
 
 
 
 {
 
 
 
 
 
 
 B
 
 
 
 c
 
 
 (
 U
 )
 ×
 
 
 O
 
 
 (
 U
 )
 →
 
 C
 
 
 
 
 
 (
 f
 ,
 φ
 )
 ↦
 ∫
 f
 ⋅
 φ
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{cases}{\mathcal {B}}_{c}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}}
 
 one associates to each hyperfunction with compact support a continuous linear function on 
 
 
 
 
 
 O
 
 
 (
 U
 )
 .
 
 
 {\displaystyle {\mathcal {O}}(U).}
 
 This induces an identification of the dual space, 
 
 
 
 
 
 
 O
 
 
 ′
 
 (
 U
 )
 ,
 
 
 {\displaystyle {\mathcal {O}}'(U),}
 
 with 
 
 
 
 
 
 
 B
 
 
 
 c
 
 
 (
 U
 )
 .
 
 
 {\displaystyle {\mathcal {B}}_{c}(U).}
 
 A special case worth considering is the case of continuous functions or distributions with compact support: If one considers 
 
 
 
 
 C
 
 c
 
 
 0
 
 
 (
 U
 )
 
 
 {\displaystyle C_{c}^{0}(U)}
 
 (or 
 
 
 
 
 
 
 E
 
 
 ′
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {E}}'(U)}
 
) as a subset of 
 
 
 
 
 
 B
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {B}}(U)}
 
 via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If 
 
 
 
 u
 ∈
 
 
 
 E
 
 
 ′
 
 (
 U
 )
 
 
 {\displaystyle u\in {\mathcal {E}}'(U)}
 
 is a distribution with compact support, 
 
 
 
 φ
 ∈
 
 
 O
 
 
 (
 U
 )
 
 
 {\displaystyle \varphi \in {\mathcal {O}}(U)}
 
 is a real analytic function, and 
 
 
 
 supp
 ⁡
 (
 u
 )
 ⊂
 (
 a
 ,
 b
 )
 
 
 {\displaystyle \operatorname {supp} (u)\subset (a,b)}
 
 then 
 
 
 
 
 ∫
 
 a
 
 
 b
 
 
 u
 ⋅
 φ
 =
 ⟨
 u
 ,
 φ
 ⟩
 .
 
 
 {\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}
 
 Thus this notion of integration gives a precise meaning to formal expressions like 
 
 
 
 
 ∫
 
 a
 
 
 b
 
 
 δ
 (
 x
 )
 
 d
 x
 
 
 {\displaystyle \int _{a}^{b}\delta (x)\,dx}
 
 which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in 
 
 
 
 
 
 E
 
 
 (
 U
 )
 ,
 
 
 
 E
 
 
 ′
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}
 
 is a subspace of 
 
 
 
 
 
 
 O
 
 
 ′
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {O}}'(U)}
 
. This is an alternative description of the same embedding 
 
 
 
 
 
 
 E
 
 
 ′
 
 ↪
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}
 
.</li>
<li>If 
 
 
 
 Φ
 :
 U
 →
 V
 
 
 {\displaystyle \Phi :U\to V}
 
 is a real analytic map between open sets of 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
, then composition with 
 
 
 
 Φ
 
 
 {\displaystyle \Phi }
 
 is a well-defined operator from 
 
 
 
 
 
 B
 
 
 (
 V
 )
 
 
 {\displaystyle {\mathcal {B}}(V)}
 
 to 
 
 
 
 
 
 B
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {B}}(U)}
 
: 
 
 
 
 f
 ∘
 Φ
 :=
 (
 
 f
 
 +
 
 
 ∘
 Φ
 ,
 
 f
 
 −
 
 
 ∘
 Φ
 )
 
 
 {\displaystyle f\circ \Phi :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}
 
</li></ul>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Algebraic_analysis/yBSslC4H">Algebraic analysis</a></li>
<li><a href="/facts/Generalized_function/fCM4VsyT">Generalized function</a></li>
<li><a href="/facts/Distribution_(mathematics)/yfaJ4mUp">Distribution (mathematics)</a></li>
<li><a href="/facts/Microlocal_analysis/rN65j9zV">Microlocal analysis</a></li>
<li><a href="/facts/Pseudo-differential_operator/wEPLVm5c">Pseudo-differential operator</a></li>
<li><a href="/facts/Sheaf_cohomology/TrcpmD6k">Sheaf cohomology</a></li></ul>

<ul><li><a href="/facts/Isao_Imai_(physicist)/k6hd51Vq">Imai, Isao</a> (2012) [1992], <a href="https://www.springer.com/book/9780792315070">Applied Hyperfunction Theory</a>, Mathematics and its Applications (Book 8), Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-94-010-5125-5.</li>
<li>Kaneko, Akira (1988), <a 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="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-90-277-2837-1</li>
<li><a href="/facts/Masaki_Kashiwara/29L71Lgt">Kashiwara, Masaki</a>; Kawai, Takahiro; Kimura, Tatsuo (2017) [1986], <a 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="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-62832-5</li>
<li>Komatsu, Hikosaburo, ed. (1973), <a 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="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-06218-9.
<ul><li>Komatsu, Hikosaburo, Relative cohomology of sheaves of solutions of differential equations, pp. 192–261.</li>
<li>Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki, Microfunctions and pseudo-differential equations, pp. 265–529. - It is called SKK.</li></ul></li>
<li><a href="/facts/Andr%25C3%25A9_Martineau/WSE7Jrjs">Martineau, André</a> (1960–1961), <a 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="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1611794">1611794</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0122.34902">0122.34902</a>.</li>
<li>Morimoto, Mitsuo (1993), <a href="https://archive.org/details/introductiontosa0000mori">An Introduction to Sato's Hyperfunctions</a>, Translations of Mathematical Monographs (Book 129), American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-82184571-4.</li>
<li>Pham, F. L., ed. (1975), <a 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="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-37454-1.
<ul><li>Cerezo, A.; Piriou, A.; Chazarain, J., Introduction aux hyperfonctions, pp. 1–53.</li></ul></li>
<li><a href="/facts/Mikio_Sato/LUkFh98o">Sato, Mikio</a> (1958), <a 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="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.11429%2Fsugaku1947.10.1">10.11429/sugaku1947.10.1</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0039-470X">0039-470X</a></li>
<li>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="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2261%2F6027">2261/6027</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0114124">0114124</a>.</li>
<li>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="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2261%2F6031">2261/6031</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0132392">0132392</a>.</li>
<li><a href="/facts/Pierre_Schapira_(mathematician)/AY5XNkvQ">Schapira, Pierre</a> (1970), <a href="https://www.springer.com/book/9783540049159">Theories des Hyperfonctions</a>, Lecture Notes in Mathematics 126, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-04915-9.</li>
<li>Schlichtkrull, Henrik (2013) [1984], <a 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="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-9775-8</li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Jacobs, Bryan. <a href="https://mathworld.wolfram.com/Hyperfunction.html">"Hyperfunction"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li>Kaneko, A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hyperfunction">"Hyperfunction"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

Hyperfunction open-in-new

Hyperfunction