Hyperfunction

A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper half-plane 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 ${\displaystyle f-g}$ 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 complex plane, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.

• 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).
• The Heaviside step function can be represented as
  $${\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log(z),-{\frac {1}{2\pi i}}\log(z)\right).}$$

  
  where ${\displaystyle \log(z)}$ is the principal value of the complex logarithm of z.
• The Dirac delta "function" is represented by
  $${\displaystyle \left({\tfrac {1}{2\pi iz}},{\tfrac {1}{2\pi iz}}\right).}$$

  
  This is really a restatement of Cauchy's integral formula. 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.
• If g is a continuous function (or more generally a distribution) 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
  $${\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.}$$

  
  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 convolution of itself with the Dirac delta function.
• 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 ${\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}$
• If f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e^1/z), then ${\displaystyle (f,-f)}$ is a hyperfunction with support 0 that is not a distribution. If f has a pole of finite order at 0 then ${\displaystyle (f,-f)}$ is a distribution, so when f has an essential singularity then ${\displaystyle (f,-f)}$ looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)

Let ${\displaystyle U\subseteq \mathbb {R} }$ be any open subset.

• By definition ${\displaystyle {\mathcal {B}}(U)}$ is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly:
  $${\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}$$

  
• The obvious restriction maps turn ${\displaystyle {\mathcal {B}}}$ into a sheaf (which is in fact flabby).
• Multiplication with real analytic functions ${\displaystyle h\in {\mathcal {O}}(U)}$ and differentiation are well-defined:
  $${\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\[6pt]{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}}$$

  
  With these definitions ${\displaystyle {\mathcal {B}}(U)}$ becomes a D-module and the embedding ${\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}$ is a morphism of D-modules.
• A point ${\displaystyle a\in U}$ is called a holomorphic point of ${\displaystyle f\in {\mathcal {B}}(U)}$ if ${\displaystyle f}$ restricts to a real analytic function in some small neighbourhood of ${\displaystyle a.}$ If ${\displaystyle a\leqslant b}$ are two holomorphic points, then integration is well-defined:
  $${\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz}$$

  
  where ${\displaystyle \gamma _{\pm }:[0,1]\to \mathbb {C} ^{\pm }}$ are arbitrary curves with ${\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}$ The integrals are independent of the choice of these curves because the upper and lower half plane are simply connected.
• Let ${\displaystyle {\mathcal {B}}_{c}(U)}$ be the space of hyperfunctions with compact support. Via the bilinear form
  $${\displaystyle {\begin{cases}{\mathcal {B}}_{c}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}}$$

  
  one associates to each hyperfunction with compact support a continuous linear function on ${\displaystyle {\mathcal {O}}(U).}$ This induces an identification of the dual space, ${\displaystyle {\mathcal {O}}'(U),}$ with ${\displaystyle {\mathcal {B}}_{c}(U).}$ A special case worth considering is the case of continuous functions or distributions with compact support: If one considers ${\displaystyle C_{c}^{0}(U)}$ (or ${\displaystyle {\mathcal {E}}'(U)}$) as a subset of ${\displaystyle {\mathcal {B}}(U)}$ via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If ${\displaystyle u\in {\mathcal {E}}'(U)}$ is a distribution with compact support, ${\displaystyle \varphi \in {\mathcal {O}}(U)}$ is a real analytic function, and ${\displaystyle \operatorname {supp} (u)\subset (a,b)}$ then
  $${\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}$$

  
  Thus this notion of integration gives a precise meaning to formal expressions like
  $${\displaystyle \int _{a}^{b}\delta (x)\,dx}$$

  
  which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in ${\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}$ is a subspace of ${\displaystyle {\mathcal {O}}'(U)}$. This is an alternative description of the same embedding ${\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}$.
• If ${\displaystyle \Phi :U\to V}$ is a real analytic map between open sets of ${\displaystyle \mathbb {R} }$, then composition with ${\displaystyle \Phi }$ is a well-defined operator from ${\displaystyle {\mathcal {B}}(V)}$ to ${\displaystyle {\mathcal {B}}(U)}$:
  $${\displaystyle f\circ \Phi$$ :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}

  

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