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 [math]\displaystyle{ f_i }[/math], its limit [math]\displaystyle{ f }[/math] is the distribution given by
- [math]\displaystyle{ f[\varphi] = \lim_{i \to \infty} f_i[\varphi] }[/math]
for each test function [math]\displaystyle{ \varphi }[/math], provided that distribution exists. The existence of the limit [math]\displaystyle{ f }[/math] means that (1) for each [math]\displaystyle{ \varphi }[/math], the limit of the sequence of numbers [math]\displaystyle{ f_i[\varphi] }[/math] exists and that (2) the linear functional [math]\displaystyle{ f }[/math] 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:
- [math]\displaystyle{ f_t(x) = {t \over 1 + t^2 x^2} }[/math]
Since, by integration by parts,
- [math]\displaystyle{ \langle f_t, \phi \rangle = -\int_{-\infty}^0 \arctan(tx) \phi'(x) \, dx - \int_0^\infty \arctan(tx) \phi'(x) \, dx, }[/math]
we have: [math]\displaystyle{ \displaystyle \lim_{t \to \infty} \langle f_t, \phi \rangle = \langle \pi \delta_0, \phi \rangle }[/math]. That is, the limit of [math]\displaystyle{ f_t }[/math] as [math]\displaystyle{ t \to \infty }[/math] is [math]\displaystyle{ \pi \delta_0 }[/math].
Let [math]\displaystyle{ f(x+i0) }[/math] denote the distributional limit of [math]\displaystyle{ f(x+iy) }[/math] as [math]\displaystyle{ y \to 0^+ }[/math], if it exists. The distribution [math]\displaystyle{ f(x-i0) }[/math] is defined similarly.
One has
- [math]\displaystyle{ (x - i 0)^{-1} - (x + i 0)^{-1} = 2 \pi i \delta_0. }[/math]
Let [math]\displaystyle{ \Gamma_N = [-N-1/2, N+1/2]^2 }[/math] be the rectangle with positive orientation, with an integer N. By the residue formula,
- [math]\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). }[/math]
On the other hand,
- [math]\displaystyle{ \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} }[/math]
Oscillatory integral
See also
References
- Demailly, Complex Analytic and Differential Geometry
- Hörmander, Lars, The Analysis of Linear Partial Differential Operators, Springer-Verlag
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