Articles of distribution theory

Is distribution theory necessary for most users of the Dirac delta function?

What would be wrong with defining the Dirac delta function as $$ \int_{-\infty}^{\infty} \delta^{(n)}(f(x)) g(x) \,dx := \lim_{h\rightarrow 0}\int_{-\infty}^{\infty} \delta_h^{(n)}(f(x)) g(x) \,dx $$ for a suitable nascent delta function $\delta_h(x)$? For example, the rectangular pulse, hat function, and normal distribution nascent delta functions are all suitable in the case $n=0.$ For $n\leq1,$ the hat function […]

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I’m studying Quillen’s rational homotopy theory and trying to understand this MathOverflow description of Quillen’s functor provided by Hiro Lee Tanaka. When discussing connections between how algebraists might find Lie algebras (as the primitives of Hopf algebras) and how geometers might find Lie algebras (as the tangent space at identity of a Lie group) Tanaka […]

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the localized Sobolev space $H_{s}^{loc}$ is the set of all distributions $f\in \mathcal{D’}(U)$ such that for every precompact open set $V$ with $\overline{V}\subset U$ there exists $g\in H_{s}$ such that $g=f$ on $V.$ Fact. A […]

Convolution and Dirac delta

I need to prove the following: $\delta_0 * \phi = \phi$, where $\phi$ is a test function. Thank you for your help.

Identify distribution by a constant function

Possible Duplicate: On distributions over $\mathbb R$ whose derivatives vanishes Why can I identify a distribution $G \in \mathcal{D}'((a,b))$, $\partial G = 0$ by a constant function?

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that $\Phi'(u)\in W_0^{1,2}(\Omega)$. Is it true that $\Delta (\Phi\circ u)$ is a measure and $$\Delta (\Phi\circ u)=(\Phi”\circ u)|\nabla u|^2+(\Phi’\circ u)\Delta u, \tag{1}$$ If the equality $$\int_\Omega \phi […]

Simplifying the generalized function $x^{\lambda}_+$ in the strip $-n – 1 < \mbox{Re}\lambda < -n$

Note: this post is a follow up to an earlier question. The (divergent) integral of $x^{\lambda}_+$ can be analytically continued into the region Re $\lambda > -n – 1$, $\lambda \ne -1, -2 , \ldots , -n$ by the expression $$ \begin{align*} \int_0^\infty x^{\lambda} \phi \: dx &= \int_0^1 x^{\lambda} \left[ \phi(x) – \phi(0) – […]

Distributional derivative of absolute value function

I’m tying to understand distributional derivatives. That’s why I’m trying to calculate the distributional derivative of $|x|$, but I got a little confused. I know that a weak derivative would be $\operatorname{sgn}(x)$, but not only I’m not finding that one in my calculations, I ended up with a wrong solution and in my other attempt […]

Delta function in curvilinear coordinates

I have been looking everywhere but I am unable to prove $$\delta(\vec{x}-\vec{a}) = \frac{1}{fgh}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$ Where $f,g,h$ are scale factors for an orthogonal system $u,v,w$. If $\vec{a}$ lies on a degenerate coordinate then $$\delta(\vec{x}-\vec{a}) = \frac{1}{fg\int hdw}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$ I know that the delta function is a generalized function, and is generally used in the form […]

Is $|x|^{-r}$ tempered distribution?

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |(1+|x|)^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ $\|f\|_{(\alpha, \beta)}:= \sup_{x\in \mathbb R} |(1+|x|)^{\alpha} D^{\beta}f(x)|$ Put $g(x)= \frac{1}{|x|^{r}}, (r>0, x\in \mathbb R).$ My Question is: Can we expect to find constant $C$ and $\alpha$ such that $$|\int_{\mathbb R} […]