Generalized functions

pseudo-function δ such that an integral of δ(x-c)f(x) always takes the value of f(c)

discontinuous function whose value is zero for negative numbers and one for positive numbers

Green's functions

a continuous functional on a space of test functions (Schwartz space), which generalizes the concept of locally integrable functions

value that can be assigned to certain divergent integrals over a finite interval

theorem

weak derivation

periodic distribution ("function") of "point-mass" Dirac delta sampling

operator on functions, defined by the composition of Fourier transformation, multiplication with a certain smooth function of both position and momentum, and inverse Fourier transformation

solutions of a certain class of inhomogeneous partial differential equations

mathematical theorem on recovering a function from its Fourier transform

generalizations of mathematical functions, elements of a space that extends a function space

any of a family of theorems relating decay properties of a function or distribution at infinity with analyticity of its Fourier transform

mathematical solution

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.

technique of studying linear partial differential equations

an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform

branch of analysis focusing on localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point under the Fourier transform

contnuous linear functional on the space of compactly supported differential forms

Hilbert space equipped with a dense subspace that carries (in addition to the subspace topology) a finer topology such that the inclusion map is continuous