Inmathematics, specifically in the theory ofgeneralized functions, thelimit 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 alimit 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 ofcalculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept offunctions.

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

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

for each test function${\displaystyle \phi }$, provided that distribution exists. The existence of the limit${\displaystyle f}$ means that (1) for each${\displaystyle \phi }$, thelimit of the sequence of numbers${\displaystyle f_i[\phi ]}$ exists and that (2) the linear functional${\displaystyle f}$ 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 theclassical limit does not. Consider, for example, the function:

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

Since, byintegration by parts,

${\displaystyle ⟨f_t,\varphi ⟩=-{\int }_{-\mathrm{\infty }}^0\arctan (tx){\varphi }^{\prime }(x)dx-{\int }_0^{\mathrm{\infty }}\arctan (tx){\varphi }^{\prime }(x)dx,}$

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

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

One has

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

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

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

On the other hand,

• Distribution (number theory)

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

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