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.

Definition
Given a sequence of distributions $$f_i$$, its limit $$f$$ is the distribution given by
 * $$f[\varphi] = \lim_{i \to \infty} f_i[\varphi]$$

for each test function $$\varphi$$, provided that distribution exists. The existence of the limit $$f$$ means that (1) for each $$\varphi$$, the limit of the sequence of numbers $$f_i[\varphi]$$ exists and that (2) the linear functional $$f$$ 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.

Examples
A distributional limit may still exist when the classical limit does not. Consider, for example, the function:
 * $$f_t(x) = {t \over 1 + t^2 x^2}$$

Since, by integration by parts,
 * $$\langle f_t, \phi \rangle = -\int_{-\infty}^0 \arctan(tx) \phi'(x) \, dx - \int_0^\infty \arctan(tx) \phi'(x) \, dx,$$

we have: $$\displaystyle \lim_{t \to \infty} \langle f_t, \phi \rangle = \langle \pi \delta_0, \phi \rangle$$. That is, the limit of $$f_t$$ as $$t \to \infty$$ is $$\pi \delta_0$$.

Let $$f(x+i0)$$ denote the distributional limit of $$f(x+iy)$$ as $$y \to 0^+$$, if it exists. The distribution $$f(x-i0)$$ is defined similarly.

One has
 * $$(x - i 0)^{-1} - (x + i 0)^{-1} = 2 \pi i \delta_0.$$

Let $$\Gamma_N = [-N-1/2, N+1/2]^2$$ be the rectangle with positive orientation, with an integer N. By the residue formula,
 * $$I_N \overset{\mathrm{def}} = \int_{\Gamma_N} \widehat{\phi}(z) \pi \cot(\pi z) \, dz = {2 \pi i} \sum_{-N}^N \widehat{\phi}(n).$$

On the other hand,
 * $$\begin{align} \int_{-R}^R \widehat{\phi}(\xi) \pi \operatorname{cot}(\pi \xi) \, d &= \int_{-R}^R \int_0^\infty \phi(x)e^{-2 \pi I x \xi} \, dx \, d\xi + \int_{-R}^R \int_{-\infty}^0 \phi(x)e^{-2 \pi I x \xi} \, dx \, d\xi \\

&= \langle \phi, \cot(\cdot - i0) - \cot(\cdot - i0) \rangle \end{align}$$