Hyperfunction

In, hyperfunctions are generalizations of functions, as a 'jump' from one to another at a boundary, and can be thought of informally as s of infinite order. Hyperfunctions were introduced by in  in Japanese, (,  in English), building upon earlier work by,  and others.

Formulation
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 $$f -g$$ 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, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.

Definition in one dimension
The motivation can be concretely implemented using ideas from. Let $$\mathcal{O}$$ be the of s on $$\Complex.$$ Define the hyperfunctions on the  as the first  group:


 * $$\mathcal{B}(\R) = H^1_{\R}(\Complex, \mathcal{O}).$$

Concretely, let $$\Complex^+$$ and $$\Complex^-$$ be the and  respectively. Then $$\Complex^+ \cup \Complex^- = \Complex \setminus \R$$ so


 * $$H^1_{\R }(\Complex, \mathcal{O}) = \left [ H^0(\Complex ^+, \mathcal{O}) \oplus H^0(\Complex ^-, \mathcal{O}) \right ] /H^0(\Complex , \mathcal{O}).$$

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

Examples

 * 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, &minus;f).
 * The can be represented as
 * $$H(x) = \left(-\tfrac{1}{2\pi i}\log(z),-\tfrac{1}{2\pi i}\log(z)-1\right).$$


 * The is represented by
 * $$\left(\tfrac{1}{2\pi iz},\tfrac{1}{2\pi iz}\right).$$
 * This is really a restatement of . 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 (or more generally a ) on the real line with support contained in a bounded interval I, then g corresponds to the hyperfunction (f, &minus;f), where f is a holomorphic function on the complement of I defined by
 * $$ f(z)= \frac 1 {2\pi i} \int_{x\in I} g(x) \frac 1 {z-x} \, dx.$$
 * 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  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 $$\textstyle\mathcal{D}'(\R)\to\mathcal{B}(\R).$$
 * If f is any function that is holomorphic everywhere except for an at 0 (for example, e1/z), then $$(f, -f)$$ is a hyperfunction with  0 that is not a distribution. If f has a pole of finite order at 0 then $$(f, -f)$$ is a distribution, so when f has an essential singularity then $$(f, -f)$$ looks like a "distribution of infinite order" at 0. (Note that distributions always have finite order at any point.)

Operations on hyperfunctions
Let $$U\subseteq\R$$ be any open subset.


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


 * The obvious restriction maps turn $$ \mathcal{B}$$ into a (which is in fact ).
 * Multiplication with real analytic functions $$h\in\mathcal{O}(U)$$ and differentiation are well-defined:
 * $$\begin{align}

h(f_+,f_-) &:= (hf_+, hf_-) \\ [6pt] \frac{d}{dz}(f_+,f_-) &:= \left (\frac{df_+}{dz},\frac{df_-}{dz} \right) \end{align}$$
 * With these definitions $$\mathcal{B}(U)$$ becomes a and the embedding $$\mathcal{D}'\hookrightarrow\mathcal{B}$$ is a morphism of D-modules.


 * A point $$a\in U$$ is called a holomorphic point of $$f\in\mathcal{B}(U)$$ if $$f$$ restricts to a real analytic function in some small neighbourhood of $$a.$$ If $$a\leqslant b$$ are two holomorphic points, then integration is well-defined:
 * $$\int_a^b f := -\int_{\gamma_+} f_+(z) \, dz + \int_{\gamma_-} f_-(z) \, dz$$
 * where $$\gamma_{\pm}:[0,1] \to \Complex^{\pm}$$ are arbitrary curves with $$\gamma_{\pm}(0)=a, \gamma_{\pm}(1)=b.$$ The integrals are independent of the choice of these curves because the upper and lower half plane are.


 * Let $$\mathcal{B}_c(U)$$ be the space of hyperfunctions with compact support. Via the bilinear form
 * $$\begin{cases} \mathcal{B}_c(U)\times\mathcal{O}(U)\to\Complex \\

(f,\varphi)\mapsto\int f \cdot \varphi \end{cases}$$
 * one associates to each hyperfunction with compact support a continuous linear function on $$\mathcal{O}(U).$$ This induces an identification of the dual space, $$\mathcal{O}'(U),$$ with $$\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)$$ (or $$\mathcal{E}'(U)$$) as a subset of $$\mathcal{B}(U)$$ via the above embedding, then this computes exactly the traditional Lebesgue-integral. Furthermore: If $$u\in\mathcal{E}'(U)$$ is a distribution with compact support, $$\varphi\in\mathcal{O}(U)$$ is a real analytic function, and $$\operatorname{supp}(u)\subset(a,b)$$ then
 * $$\int_a^b u\cdot\varphi = \langle u,\varphi\rangle.$$
 * Thus this notion of integration gives a precise meaning to formal expressions like
 * $$\int_a^b \delta(x) \, dx$$
 * which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in $$\mathcal{E}(U), \mathcal{E}'(U)$$ is a subspace of $$\mathcal{O}'(U)$$. This is an alternative description of the same embedding $$\mathcal{E}'\hookrightarrow\mathcal{B}$$.


 * If $$\Phi:U\to V$$ is a real analytic map between open sets of $$\R$$, then composition with $$\Phi$$ is a well-defined operator from $$\mathcal{B}(V)$$ to $$\mathcal{B}(U)$$:
 * $$f\circ\Phi:=(f_+\circ\Phi,f_-\circ\Phi)$$