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Limit of distributions
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<h2 id="definition">Definition</h2> <p>Given a sequence of distributions f i {\displaystyle f_{i}} , its limit f {\displaystyle f} is the distribution given by </p> f [ φ ] = lim i → ∞ f i [ φ ] {\displaystyle f[\varphi ]=\lim _{i\to \infty }f_{i}[\varphi ]} <p>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 <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit of the sequence</a> 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. </p><p>More generally, as with functions, one can also consider a limit of a family of distributions. </p> <h2 id="examples">Examples</h2> <p>A distributional limit may still exist when the <a href="/facts/Classical_limit/Yu7aHtr6">classical limit</a> does not. Consider, for example, the function: </p> f t ( x ) = t 1 + t 2 x 2 {\displaystyle f_{t}(x)={t \over 1+t^{2}x^{2}}} <p>Since, by <a href="/facts/Integration_by_parts/aV6JgBNd">integration by parts</a>, </p> ⟨ 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,} <p>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}} . </p><p>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. </p><p>One has </p> ( 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}.} <p>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 <i>N</i>. By the <a href="/facts/Residue_formula/M5ITFIyx">residue formula</a>, </p> 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).} <p>On the other hand, </p> ∫ − 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}}} <h2 id="oscillatory-integral">Oscillatory integral</h2> <p class="note">Main article: <a href="/facts/Oscillatory_integral/g3EC35jg">Oscillatory integral</a></p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Distribution_(number_theory)/UNrPIzXV">Distribution (number theory)</a></li></ul> <ul><li>Demailly, Complex Analytic and Differential Geometry</li> <li><a href="/facts/Lars_H%25C3%25B6rmander/mxyNnmRA">Hörmander</a>, Lars, <i>The Analysis of Linear Partial Differential Operators</i>, Springer-Verlag</li></ul>