In 1935, Torsten Carleman introduced a generalisation of Fourier transform, which foreshadowed the work of Mikio Sato on hyperfunctions; his notes were published in . He considered the functions f of at most polynomial growth, and showed that every such function can be decomposed as f = f+ + f−, where f+ and f− are analytic in the upper and lower half planes, respectively, and that this representation is essentially unique.
In 1935, Torsten Carleman generalized Fourier transform, a concept that foreshadowed Mikio Sato's work on hyperfunctions. His notes were published in . He focused on functions f with at most polynomial growth and demonstrated that every such function can be decomposed into two analytic functions: f+ for the upper half plane and f− for the lower half plane. This decomposition is unique.