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Let $\phi:\mathbb R \to \mathbb R$ be a test function. We denote $D(\mathbb R)$ the set of test functions. The dirac distribution $$\delta :D(\mathbb R)\to \mathbb R$$ is defined by: $$<\delta , \phi>=\phi(0)$$ Now take any smooth function $g:\mathbb R\to \mathbb R$, and define the product distribution $g.\delta$ by: $$<g.\delta,\phi>=<\delta,g\phi>$$ From this definition we see that $$<g.\delta,\phi>=g(0)\phi(0)$$

Now i'm reading the following problem prove that $x\delta(x)=0$ and $x\delta'(x)=-\delta(x)$ wihtout any firther information, and I want to understand the meaning of these formulas regarding the definitions above, I mean what is $x$ and what is $\delta(x)$ and what is the product $x\delta(x)$? It seems like he takes $g$ for being the identity $id :\mathbb R\to \mathbb R$ but this would give $$<id.\delta, \phi>=id(0)\phi(0)=0*\phi(0)=0$$ but i don't see where $x\delta(x)=0$ come from. thank you for your help!

palio
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  • From definition $<x.\delta,\phi>=x(0)\phi(0)=0$. Here $x$ is the identity function that maps $y$ to $y$ and hence $0$ to $0$. – A.S. Nov 02 '15 at 10:05
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    Thank you @A.S. but you are rewriting my statements, my question is about the meaning of the expression $x\delta(x)=0$ what is $x$ and what is $\delta(x)$ and what is the product $x\delta(x)$ ? – palio Nov 02 '15 at 10:25
  • The meaning of $x\delta(x)=0$ is that if you integrate $x\delta(x)$ against any test function you will get $0$, hence $x\delta(x)=0$. Let me ask you - what is the meaning of $\delta(x)$? – A.S. Nov 02 '15 at 10:28
  • \begin{eqnarray} \int dxx\delta ^{\prime }(x)g(x) &=&\left[ x\delta (x)g(x)\right] _{x=-\infty }^{+\infty }-\int dx\delta (x)\partial _{x}{xg(x)} \ &=&-\int dx\delta (x)g(x)-\int dx\delta (x)x\partial _{x}g(x) \ &=&-\int dx\delta (x)g(x) \end{eqnarray} – Urgje Nov 02 '15 at 10:36
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    I believe you should just read $x \delta(x)$ as $\text{id}_{\mathbb{R}}.\delta$. Some people think of distributions as though they were functions on $\mathbb{R}$ (and then $x\delta(x)$ would make sense) only that you are not allowed to evaluate them directly at any point but integrate them against test functions (read: evaluate at). – Matthias Klupsch Nov 02 '15 at 10:36
  • @A.S. That's exactly my question, what is $\delta(x)$ as the definition I gave is that $\delta$ maps a test function $\phi$ to the real number $\phi(0)$ so i really don't understand what is $\delta(x)$ !! – palio Nov 02 '15 at 10:47
  • @MatthiasKlupsch So you are saying that $x\delta(x)=0$ is just a rough way of writing that the product distribution $id_{\mathbb R}.\delta$ is just the zero distribution that maps any test function to zero ? If this is what you mean then could you please explain what do you mean by "integrate them against test functions" ? – palio Nov 02 '15 at 10:57
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    Yes, that is exactly what I meant. In the context where you interpret a distribution as a function, evaluation $\langle f , g \rangle$ of $f$ at $g$ is interpreted as integrating the product $f g$ over $\mathbb{R}$. See the comment by Urgje for an example. – Matthias Klupsch Nov 02 '15 at 12:04
  • Now I see what you mean, thank you ! – palio Nov 02 '15 at 13:33

2 Answers2

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  1. For a distribution $u$ and polynomials $P,Q$, we define the distribution $P(M_{{\rm id}_{\mathbb{R}}}) Q(\partial)u$ as $$\tag{1} \left(P(M_{{\rm id}_{\mathbb{R}}}) Q(\partial)u\right) [\varphi]~:=~ u\left[ Q(-\partial)P(M_{{\rm id}_{\mathbb{R}}})\varphi\right], $$ where $\varphi$ is a test function; where $M_f(g):=fg$ denotes the multiplication operator with the function $f$; where $\partial(f)=f^{\prime}$ is differentiation; and where ${\rm id}_{\mathbb{R}}$ is the identity function: $\mathbb{R}\to \mathbb{R}$. Note that the multiplication operator $M_{{\rm id}_{\mathbb{R}}}$ is often denoted by the indeterminate $x$ by a slight abuse of notation.

  2. The Dirac delta distribution $u=\delta$ is defined as $$\tag{2}\delta [\varphi]~:=~\varphi(0).$$

  3. The distribution identities $$\tag{3} M_{{\rm id}_{\mathbb{R}}}\delta~=~0\qquad\text{and}\qquad M_{{\rm id}_{\mathbb{R}}}\partial\delta~=~-\delta,$$ (which is often written as $$\tag{4} x\delta(x)~=~0\qquad\text{and}\qquad x\delta^{\prime}(x)~=~-\delta(x),$$ with a slight abuse of notation), follow from $$\tag{5} \left(M_{{\rm id}_{\mathbb{R}}}\delta\right)[\varphi]~\stackrel{(1)}{=}~\delta\left[M_{{\rm id}_{\mathbb{R}}} \varphi\right]~\stackrel{(2)}{=}~0\varphi(0)~=~0~=~0[\varphi],$$ and $$ \left(M_{{\rm id}_{\mathbb{R}}}\partial\delta\right)[\varphi]~\stackrel{(1)}{=}~\delta\left[-\partial M_{{\rm id}_{\mathbb{R}}} \varphi\right]~\stackrel{\text{Leibniz' rule}}{=}~\delta\left[-\varphi- M_{{\rm id}_{\mathbb{R}}} \partial\varphi\right]$$ $$\tag{6}~\stackrel{(2)}{=}~-\varphi(0)-0\varphi^{\prime}(0)~\stackrel{(2)}{=}~-\delta[\varphi],$$ respectively.

Qmechanic
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By definition we can write for any test function $\phi$ $$\langle x\delta_0,\phi(x)\rangle = \langle \delta_0, x\phi(x)\rangle = 0\cdot \phi(0)=0=\langle 0,\phi(x)\rangle,$$ which means that $x\delta_0=0$ in the sense of distributions.

In the same spirit, we write

$$\langle x\delta'_0,\phi(x)\rangle = \langle \delta_0', x\phi(x)\rangle =- \langle \delta_0,(x\phi(x))'\rangle = -\langle \delta_0,\phi(x)+x\phi'(x) \rangle = -\phi(0)-0\cdot\phi'(0) = -\phi(0) = -\langle \delta_0, \phi(x) \rangle,$$ which means that $x\delta'_0= -\delta_0$ in the sense of distributions.

TZakrevskiy
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