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.

Given a sequence of distributions ${\displaystyle f_{i}}${\displaystyle f_{i}}, its limit ${\displaystyle f}${\displaystyle f} is the distribution given by

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

for each test function ${\displaystyle \varphi }${\displaystyle \varphi }, provided that distribution exists. The existence of the limit ${\displaystyle f}${\displaystyle f} means that (1) for each ${\displaystyle \varphi }${\displaystyle \varphi }, the limit of the sequence of numbers ${\displaystyle f_{i}[\varphi ]}${\displaystyle f_{i}[\varphi ]} exists and that (2) the linear functional ${\displaystyle f}${\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.

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

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

Since, by integration by parts,

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

we have: ${\displaystyle \displaystyle \lim _{t\to \infty }\langle f_{t},\phi \rangle =\langle \pi \delta _{0},\phi \rangle }${\displaystyle \displaystyle \lim _{t\to \infty }\langle f_{t},\phi \rangle =\langle \pi \delta _{0},\phi \rangle }. That is, the limit of ${\displaystyle f_{t}}${\displaystyle f_{t}} as ${\displaystyle t\to \infty }${\displaystyle t\to \infty } is ${\displaystyle \pi \delta _{0}}${\displaystyle \pi \delta _{0}}.

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

One has

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

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

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

On the other hand,

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

• Distribution (number theory)

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

• Mathematical analysis stubs
• Generalized functions
• Schwartz distributions

• All stub articles