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 f i {\displaystyle f_{i}} , its limit f {\displaystyle f} is the distribution given by

f [ φ ] = lim i → ∞ f i [ φ ] {\displaystyle f[\varphi ]=\lim _{i\to \infty }f_{i}[\varphi ]}

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

Examples

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

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

Since, by integration by parts,

⟨ f t , ϕ ⟩ = − ∫ − ∞ 0 arctan ⁡ ( t x ) ϕ ′ ( x ) d x − ∫ 0 ∞ arctan ⁡ ( t x ) ϕ ′ ( x ) d x , {\displaystyle \langle f_{t},\phi \rangle =-\int _{-\infty }^{0}\arctan(tx)\phi '(x)\,dx-\int _{0}^{\infty }\arctan(tx)\phi '(x)\,dx,}

we have: lim t → ∞ ⟨ f t , ϕ ⟩ = ⟨ π δ 0 , ϕ ⟩ {\displaystyle \displaystyle \lim _{t\to \infty }\langle f_{t},\phi \rangle =\langle \pi \delta _{0},\phi \rangle } . That is, the limit of f t {\displaystyle f_{t}} as t → ∞ {\displaystyle t\to \infty } is π δ 0 {\displaystyle \pi \delta _{0}} .

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

One has

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

Let Γ 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,

I N = d e f ∫ Γ N ϕ ^ ( z ) π cot ⁡ ( π z ) d z = 2 π i ∑ − N N ϕ ^ ( n ) . {\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).}

On the other hand,

∫ − R R ϕ ^ ( ξ ) π cot ⁡ ( π ξ ) d = ∫ − R R ∫ 0 ∞ ϕ ( x ) e − 2 π I x ξ d x d ξ + ∫ − R R ∫ − ∞ 0 ϕ ( x ) e − 2 π I x ξ d x d ξ = ⟨ ϕ , cot ⁡ ( ⋅ − i 0 ) − cot ⁡ ( ⋅ − i 0 ) ⟩ {\displaystyle {\begin{aligned}\int _{-R}^{R}{\widehat {\phi }}(\xi )\pi \operatorname {cot} (\pi \xi )\,d&=\int _{-R}^{R}\int _{0}^{\infty }\phi (x)e^{-2\pi Ix\xi }\,dx\,d\xi +\int _{-R}^{R}\int _{-\infty }^{0}\phi (x)e^{-2\pi Ix\xi }\,dx\,d\xi \\&=\langle \phi ,\cot(\cdot -i0)-\cot(\cdot -i0)\rangle \end{aligned}}}

Oscillatory integral

See also

  • Distribution (number theory)

References

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


  • v
  • t
  • e