In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese, (1959, 1960 in English), building upon earlier work by Laurent Schwartz, Grothendieck 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 {\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.
Definition in one dimension The motivation can be concretely implemented using ideas from sheaf cohomology. Let O {\displaystyle {\mathcal {O}}} be the sheaf of holomorphic functions on C . {\displaystyle \mathbb {C} .} Define the hyperfunctions on the real line as the first local cohomology 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 upper half-plane and lower half-plane 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.
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