hyperfunktioner

Hyperfunctions are a mathematical concept used in real analysis and the theory of partial differential equations as a generalization of distributions. Informally, a hyperfunction on an open set in R^n is described as a boundary value of a holomorphic function defined on a complex neighborhood of that set. In one dimension, a hyperfunction on an interval can be thought of as the difference between the boundary values of two holomorphic functions, one defined just above and one just below the real axis. More formally, hyperfunctions form a sheaf that can be described as a first cohomology group with coefficients in the sheaf of holomorphic functions, capturing a global boundary-value perspective.

Hyperfunctions include all distributions and extend beyond them in both growth and singular behavior. They are closed under differentiation and under multiplication by real-analytic functions, and there is a compatible Fourier transform theory, including tempered variants. This makes hyperfunction theory particularly well suited to microlocal analysis and the study of linear partial differential equations, where the fine structure of singularities and their propagation is essential.

Key features and examples:

- Any distribution can be realized as a hyperfunction, but hyperfunctions also model more singular or analytically intricate boundary phenomena that distributions cannot easily capture.

- Hyperfunctions are naturally defined on real analytic manifolds and are linked to the boundary values of holomorphic functions on complexifications.

- They form a flexible framework for analytic functionals, analytic continuation, and the algebraic structure needed in D-module theory.

History and scope:

Hyperfunctions were developed in the 1950s–1960s by Mikusinski, Sato and collaborators, with the formalism leveraging sheaf theory to treat boundary values of holomorphic functions. Today they remain a fundamental tool in several complex variables, microlocal analysis, and the analytic theory of PDEs.