Limit of distributions

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.

Given a sequence of distributions ${\displaystyle f_{i}}$, its limit ${\displaystyle f}$ is the distribution given by

for each test function ${\displaystyle \varphi }$, provided that distribution exists. The existence of the limit ${\displaystyle f}$ means that (1) for each ${\displaystyle \varphi }$, the limit of the sequence of numbers ${\displaystyle f_{i}[\varphi ]}$ exists and that (2) the linear functional ${\displaystyle f}$ defined by the above formula is continuous with respect to the topology on the space of test functions.

More generally, as with functions, one can also consider a limit of a family of distributions.

A distributional limit may still exist when the classical limit does not. Consider, for example, the function:

Since, by integration by parts,

we have: ${\displaystyle \displaystyle \lim _{t\to \infty }\langle f_{t},\phi \rangle =\langle \pi \delta _{0},\phi \rangle }$. That is, the limit of ${\displaystyle f_{t}}$ as ${\displaystyle t\to \infty }$ is ${\displaystyle \pi \delta _{0}}$.