Limit of distributions explained

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} 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 ^{0} _{-inf ty} \arctan(tx) \phi'(x) dx - \int ^{inf ty} _{0} \arctan(tx) \phi'(x) dx, we have: \displaystyle \lim _{t} \langle f _{t,} \phi \rangle = \langle \pi \delta _{0,} \phi \rangle . That is, the limit of f _{t} as t \to inf ty is \pi \delta _{0} .

Let

f(x + i 0 ) denote the distributional limit of f(x + iy) as y \to 0 ^{+} , if it exists. The distribution f(x - i 0 ) 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{ def } = \int_ \widehat(z) \pi \cot(\pi z) \, dz = \sum_^N \widehat(n).On the other hand,\begin{align} \int ^{R} _{- R} \widehat{\phi}(\xi) \pi \operatorname{cot}(\pi \xi) d & = \int ^{R} _{- R} \int ^{inf ty} _{0} \phi(x)e ^{-2} dx d\xi + \int ^{R} _{- R} \int ^{0} _{-inf ty} \phi(x)e ^{-2} dx d\xi \\ & = \langle \phi, \cot( ⋅ - i 0 ) - \cot( ⋅ - i 0 ) \rangle \end{align}

Oscillatory integral

See main article: Oscillatory integral.

See also

• Distribution (number theory)

References

• Demailly, Complex Analytic and Differential Geometry