In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This notion generalizes a limit of a sequence of functions; a limit as a distribution may exist when a limit of functions does not.
The notion is a part of distributional calculus, a generalized form of calculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept of functions.
A distributional limit may still exist when the classical limit does not. Consider, for example, the function:
Since, by integration by parts,
we have: . That is, the limit of as is .
Let denote the distributional limit of as , if it exists. The distribution is defined similarly.
One has
Let be the rectangle with positive orientation, with an integer N. By the residue formula,
On the other hand,
- Demailly, Complex Analytic and Differential Geometry
- Hörmander, Lars, The Analysis of Linear Partial Differential Operators, Springer-Verlag