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 $f_i$, its limit $f$ is the distribution given by

$f[\phi ]=\underset{i\to \mathrm{\infty }}{\lim }{f}_i[\phi ]$

for each test function $\phi$, provided that distribution exists. The existence of the limit $f$ means that (1) for each $\phi$, the limit of the sequence of numbers $f_i[\phi ]$ 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:

$f_t(x)=\frac{t}{1+t^2{x}^2}$

Since, by integration by parts,

$⟨f_t,\varphi ⟩=-\underset{-\mathrm{\infty }}{\overset{0}{\int }}\arctan (tx){\varphi }^{\prime }(x)dx-\underset{0}{\overset{\mathrm{\infty }}{\int }}\arctan (tx){\varphi }^{\prime }(x)dx,$

we have: $\underset{t\to \mathrm{\infty }}{\lim }⟨f_t,\varphi ⟩=⟨\pi {\delta }_0,\varphi ⟩$. That is, the limit of $f_t$ as $t\to \mathrm{\infty }$ 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

$(x-i0{)}^{-1}-(x+i0{)}^{-1}=2\pi i{\delta }_0.$

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

$I_N\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{{\mathrm{\Gamma }}_N}{\int }\hat{\varphi }(z)\pi \cot (\pi z)dz=2\pi i\sum _{-N}^N\hat{\varphi }(n).$

On the other hand,

$\begin{array}{rl}\underset{-R}{\overset{R}{\int }}\hat{\varphi }(\xi )\pi \cot (\pi \xi )d& =\underset{-R}{\overset{R}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\varphi (x)e^{-2\pi Ix\xi }dxd\xi +\underset{-R}{\overset{R}{\int }}\underset{-\mathrm{\infty }}{\overset{0}{\int }}\varphi (x)e^{-2\pi Ix\xi }dxd\xi \\ & =⟨\varphi ,\cot (\cdot -i0)-\cot (\cdot -i0)⟩\end{array}$

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