hyperfunksjon

Hyperfunksjon, or hyperfunction, is a concept in mathematical analysis that generalizes the idea of a function beyond distributions. It was developed in the 1950s in the work of Sato and colleagues as part of a program to use complex analytic methods to describe boundary values of functions. On a real-analytic manifold or open set U in R^n, a hyperfunction can be described, in one common formulation, as a boundary value determined by a pair of holomorphic functions defined on neighborhoods of U in the complexification C^n, taken from the two sides of U. The hyperfunction is then the mismatch of these boundary values along U.

Hyperfunctions extend the notion of distributions (generalized functions) in a way that can encode more singular behavior and finer boundary phenomena. Every distribution can be viewed as a hyperfunction, but the class of hyperfunctions is larger and often better suited for problems that involve sharp boundary behavior or analytic structure. A familiar intuition is that hyperfunctions capture jumps or boundary discontinuities via analytic boundary values, for example on the real line through upper and lower half-plane limits.

From a formal standpoint, hyperfunctions form a sheaf on the underlying real-analytic space, built from holomorphic data in the complex domain. This framework integrates tools from complex analysis, cohomology, and microlocal analysis, and it provides a robust setting for boundary-value problems in partial differential equations. Hyperfunction theory has influenced areas such as the study of linear PDEs, asymptotic analysis, and the development of related notions like microfunctions.

Further reading can be found in early works introducing hyperfunctions and in subsequent treatments by Sato, Hörmander, and others in the fields of analysis and microlocal theory.