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nLab > Latest Changes: convolution product of distributions

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeNov 2nd 2017
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Author: Urs
Format: MarkdownItexstarted some minimum at _[[convolution product of distributions]]_

started some minimum at convolution product of distributions
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeNov 28th 2017
- (edited Nov 28th 2017)
- PermaLink
Author: Urs
Format: MarkdownItexadded this statement ([here](https://ncatlab.org/nlab/show/convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions) and also at [[wave front set]] [here](https://ncatlab.org/nlab/show/wavefront+set#WaveFrontSetOfCompactlySupportedDistributions)): Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two [[compactly supported distributions]]. Then the [[wave front set]] of their [[convolution of distributions]] is $$ WF(u \star v) \;=\; \left\{ (x + y, k) \;\vert\; (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u) \right\} \,. $$

added this statement (here and also at wave front set here):

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u,v \in \mathcal{E}'(\mathbb{R}^n)</annotation></semantics></math>$ be two compactly supported distributions. Then the wave front set of their convolution of distributions is
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo>⋆</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo stretchy="false">|</mo><mspace width="0.27778em"/><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mtext>and</mtext><mspace width="0.16667em"/><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> WF(u \star v) \;=\; \left\{ (x + y, k) \;\vert\; (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u) \right\} \,. </annotation></semantics></math>$
- CommentRowNumber3.
- CommentAuthornLab edit announcer
- CommentTimeNov 20th 2021
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Author: nLab edit announcer
Format: MarkdownItexI checked the hormander textbook, the definition contains an error, the function convoluted with the tempered distribution has to be smooth and with compact support. Daniele Zerbini <a href="https://ncatlab.org/nlab/revision/diff/convolution+product+of+distributions/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/convolution+product+of+distributions/7">v7</a>, <a href="https://ncatlab.org/nlab/show/convolution+product+of+distributions">current</a>

I checked the hormander textbook, the definition contains an error, the function convoluted with the tempered distribution has to be smooth and with compact support.

Daniele Zerbini

diff, v7, current

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nLab > Latest Changes: convolution product of distributions