Limit of distributions

This article is about limits of sequences of generalized functions. For limits of probability distributions, see Convergence of random variables § Convergence in distribution.

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 ${\displaystyle f_{i}}$, its limit ${\displaystyle f}$ is the distribution given by

${\displaystyle f[\varphi ]=\lim _{i\to \infty }f_{i}[\varphi ]}$

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.

Examples

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

${\displaystyle f_{t}(x)={t \over 1+t^{2}x^{2}}}$

Since, by integration by parts,

${\displaystyle \langle f_{t},\phi \rangle =-\int _{-\infty }^{0}\arctan(tx)\phi '(x)\,dx-\int _{0}^{\infty }\arctan(tx)\phi '(x)\,dx,}$

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 ${\displaystyle \pi \delta _{0}}$.

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

One has

${\displaystyle (x-i0)^{-1}-(x+i0)^{-1}=2\pi i\delta _{0}.}$

Let ${\displaystyle \Gamma _{N}=[-N-1/2,N+1/2]^{2}}$ be the rectangle with positive orientation, with an integer N. By the residue formula,

${\displaystyle I_{N}{\overset {\mathrm {def} }{=}}\int _{\Gamma _{N}}{\widehat {\phi }}(z)\pi \cot(\pi z)\,dz={2\pi i}\sum _{-N}^{N}{\widehat {\phi }}(n).}$

On the other hand,

${\displaystyle {\begin{aligned}\int _{-R}^{R}{\widehat {\phi }}(\xi )\pi \operatorname {cot} (\pi \xi )\,d&=\int _{-R}^{R}\int _{0}^{\infty }\phi (x)e^{-2\pi Ix\xi }\,dx\,d\xi +\int _{-R}^{R}\int _{-\infty }^{0}\phi (x)e^{-2\pi Ix\xi }\,dx\,d\xi \\&=\langle \phi ,\cot(\cdot -i0)-\cot(\cdot -i0)\rangle \end{aligned}}}$

Oscillatory integral

Main article: Oscillatory integral

See also

• Distribution (number theory)

References

• Demailly, Complex Analytic and Differential Geometry
• Hörmander, Lars, The Analysis of Linear Partial Differential Operators, Springer-Verlag