Limit Of A 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
:$f$varphi= \lim_ f_ivarphi/math>
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/math> 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:
:$f_t(x) =$
Since, by integration by parts,
:$\langle f_t, \phi \rangle = -\int_^0 \arctan(tx) \phi'(x) \, dx - \int_0^\infty \arctan(tx) \phi'(x) \, dx,$
we have: $\displaystyle \lim_ \langle f_t, \phi \rangle = \langle \pi \delta_0, \phi \rangle$. 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
:$(x - i 0)^ - (x + i 0)^ = 2 \pi i \delta_0.$

Let \Gamma_N = N-1/2, N+1/22 be the rectangle with positive orientation, with an integer ''N''. By the residue formula,
:$I_N \overset = \int_ \widehat(z) \pi \cot(\pi z) \, dz = \sum_^N \widehat(n).$
On the other hand,
:$\begin \int_^R \widehat(\xi) \pi \operatorname(\pi \xi) \, d &= \int_^R \int_0^\infty \phi(x)e^ \, dx \, d\xi + \int_^R \int_^0 \phi(x)e^ \, dx \, d\xi \\ &= \langle \phi, \cot(\cdot - i0) - \cot(\cdot - i0) \rangle \end$

Oscillatory integral

See also

*Distribution (number theory)

References

*Demailly, Complex Analytic and Differential Geometry
*

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Generalized functions
Schwartz distributions