Use this tag for questions about Schwartz distributions, also known as Generalised Functions. For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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54 views

What is the inverse Fourier transform of $|k|^{-\alpha}$?

What is the inverse Fourier transform, $\mathcal{F}^{-1}\{|k|^{-\alpha}\}$? I am specifically interested in the case where $1<\alpha<2$. To do this, I need to compute the integral ...
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0answers
14 views

Continuous linear operator from $S'$ to $S'$

Do you know a general class of continuous linear operators from $S'$ to $S'$ ? (where $S'$ is the space of tempered distributions, dual space of the Schwartz space) Or, formally, given $X \in S'$ and ...
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0answers
31 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
3
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0answers
25 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
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1answer
23 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
3
votes
1answer
27 views

Limit in $S' (\mathbb{R})$

Given the sequence of distributions: $$ x^3~ \sin (nx),~~n \in (\mathbb{N}) $$ How can i find the limit for $n \rightarrow \infty$? I tried with the usual substitution $y=nx$, but it leads to ...
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0answers
17 views

Criteria to prove that a map is a tempered distribution

There is any simple sufficient condition to determine if a function is a tempered distribution? For example, given the map : $$ F \phi = \int_\epsilon^\infty \! \frac {\phi(x)}{\sqrt{x}} \, ...
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2answers
54 views

Check: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. [closed]

Part A: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. Part B: Generalize the previous problem and deduce a formula for the Fourier transform of a polynomial of degree m. ...
2
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0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
3
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1answer
39 views

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for ...
2
votes
1answer
48 views

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 ...
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0answers
25 views

Taylor theorem on Sobolev spaces

I am trying to understand the Taylor theorem for Sobolev spaces that appears in http://science.org.ge/moambe/5-2/5-10%20Boyarsky.pdf. I am not sure in what sense the aproximation is. I feel that it is ...
2
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1answer
29 views

Distributions with support of the form $\left\lbrace x \right\rbrace$

Doing some calculations with Distributions I came up with the following theorem: THEOREM: Let $O \subseteq \mathbb{R^d}$ be an open subset and $x \in O$. Suppose $T \in \mathcal{D}'(O)$ with ...
2
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1answer
45 views

Verifying that $\lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x)$

I'd like to show that: $$ \lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x). $$
3
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1answer
35 views

When does the regularization of a function converges to the function?

Let $\theta(x)$ equal $k\exp(-\frac{1}{1-||x||} )$ if $||x||<1$, and equal 0 if $||x||\geq1.$ Here $||.||$ designates the Euclidian norm in $\mathbb{R}^{^{n}}$, and the constant $k$ is chosen such ...
0
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2answers
44 views

Let $\langle S, \psi \rangle=\sum_{n \in \mathbb N} \int_0^n \psi'(x)dx$. Is S a distribution?

Let $\langle S, \psi\rangle=\sum_{n \in N} \int_0^n \psi'(x)dx$. Is S a distribution? I claim that S is not a distribution. I know that if S was a distribution it would satisfy the following ...
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1answer
18 views

Example of pseudodifferential operators that smooth out the singularity of delta function

What is one example of pseudodifferential operator $P$ that smooth out the singularity of delta function, i.e. $P$ s.t. $P \delta(x) \in C^{\infty}(\mathcal{R})$?
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2answers
47 views

Show that T does not have a finite order

Part A: Show that $\langle T, \psi\rangle=\sum_{n=1}^\infty \psi^{(n)}(n)$ defines a distribution. Please check: $$|\langle T, \psi\rangle|=|\sum_{n=1}^\infty \psi^{(n)}(n)| \leq \sum_{n=1}^\infty ...
4
votes
1answer
72 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
2
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1answer
45 views

Fourier transform of unit step function

It is well known that the fourier transform for unit step $U(t)$ is \begin{equation} F(U(t))=\frac{1}{j\omega}+\pi \delta(\omega) \end{equation} When I try to arrive to this expression from the ...
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1answer
32 views

Null space of $\mathrm{D_x} \mathrm{D_y}$ in $\mathcal{D}'(\mathbb{R}^2)$?

I am interested about the null space of the operator $\mathrm{D_x} \mathrm{D_y}$ on the space $\mathcal{D}'(\mathbb{R}^2)$ of generalized functions (or distributions) of Schwartz, i.e. $$\{ f \in ...
0
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1answer
38 views

Line integral along an implicit curve and dirac distibution

Let $\varphi : \Bbb{R}^2 \rightarrow \Bbb{R}$ defining an implicite curve $C = \{ (x,y), \varphi(x,y) = 0 \}$, and $u : \Bbb{R}^2 : \rightarrow \Bbb{R}$ Does the line integral $\int_C u(x,y)\ dC$ ...
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1answer
56 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
0
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1answer
32 views

partial derivatives of Dirac functions

I was reading my courses, and couldn't understand an exercise: The question was: simplify in $D'(\Bbb R^n)$ $\sum_{i=1}^n x_i\frac{\partial \delta}{\partial{x_i}}$ on my correction, I had written: ...
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2answers
52 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
4
votes
1answer
42 views

How to show distribution $T$ is a constant

Let $T$ be a distribution on $\mathbb{R}$. $\tau_a\phi(x)=\phi(x-a)$ and $\langle T,\tau_a\phi\rangle=\langle T,\phi\rangle$ for all $a\in \mathbb{R}$ and all test functions $\phi$. Prove that $T$ ...
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1answer
13 views

Do 3-D vectors of distributions (specifically vectors containing delta functions) have Helmholtz decompositions?

Define the function $f_i:\mathbb{R}^3\rightarrow\mathbb{R}^3$, $i\in\{1,2,3\}$, by $f_i(\boldsymbol{x})=\delta(\boldsymbol{x-x_0})\boldsymbol{e}_i$ where $\delta$ is the Dirac Delta function and ...
2
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1answer
96 views

Dirac delta distribution and measure?

Of course the Dirac delta is not a function. Despite, I think the concept of a measure is much easier than that of a distribution. Therefore, I was wondering: In what sense is the concept of a Dirac ...
3
votes
2answers
58 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
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1answer
49 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
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1answer
39 views

Prove or disprove: $e^{-nG(x)}$, normalized, is an approximation to the identity for $G(x)$ strictly convex

We are given the sequence of functions $$ \phi_{n} = \frac{e^{-nG(x)}}{\int_{\mathbb{R}}e^{-nG(x)}dx}$$ for a nonnegative, strictly convex function $G$ (that is, $G'' \geq c$ for some $c>0$) that ...
1
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1answer
19 views

Cauchy singular integral operator

Help on proving the following equality: $$K(-sgn)=S$$ where $K$ is the operator defined by $K(f)=F^{−1}fF$ ($F$=fourier transform, $f$=any function), sgn is the signum function and S is the Cauchy ...
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1answer
20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
4
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0answers
62 views

Limit of distribution

Let $T\in\mathcal{D}'(\mathbb{R})$ be a distribution on the set of smooth functions of compact support $\mathcal{D}(\mathbb{R})$ such that $$ \forall_{g\in\mathcal{D}(\mathbb{R})}~|\langle T, g ...
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0answers
45 views

p--laplacian problem

For $f \in W^{-1,p'}(\Omega)$, $f \geq 0$, we definr $v,$ $v_{\epsilon}$, $g_{\epsilon}$ by $$-\mathrm{div}(|Dv|^{p-2} Dv)=f \quad \mbox{in} \mathcal{D}'(\Omega), \quad v \in W^{1,p}_0(\Omega)$$ with ...
0
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1answer
43 views

Surface area of sphere in N dimensions; and a failed extension to ellipsoids

I'll present a calculation of the surface area of a sphere in $N$-dimensions. This calculation is performed in cartesian coordinates. I haven't seen the computation done this way before (though I ...
2
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1answer
46 views

Weak-Strong Derivatives

If a continuous function $u:\mathbb R^d\to \mathbb R$ has a weak derivative $\frac{\partial u}{\partial x_j}$ that exists everywhere as a locally integrable function, and it is even continuous, does ...
4
votes
1answer
51 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
2
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1answer
36 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...
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1answer
33 views

Limit of $t^2 \cos t x$ as distribution.

I am interested in finding the limit $t^2 \cos t x$ as $t \rightarrow \infty$ in the sense of schwartz distributions. After some integration by parts I get $$( t^2 \cos t x,\phi)=\int ...
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0answers
46 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
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0answers
33 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
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1answer
56 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
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0answers
33 views

Finding weak solutions

I am trying to understand how to solve differential equations of distributions. For example if one consider $ u' + u = \delta_{0}$, where $ u \in \mathcal{D}'$, this would correspond to $<u, ...
2
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1answer
31 views

Duals of embeddings in the space of distributions

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of ...
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1answer
57 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
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1answer
70 views

eigenvalues to Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
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0answers
36 views

When can we interchange summation with $L^2$ inner product?

(This question concerns a step in the solution given to Eignvalues of Laplacian operator and Sobolev spaces.) Why can we interchange the sum and the $L^2$ inner product in the following? $$(\sum_n ...
3
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0answers
40 views

Do we have $C^\infty \cap \mathcal{O}_C' = \mathcal{S}$ and/or $C^\infty \cap \mathcal{S}' = \mathcal{O}_M$?

We define the following traditional function spaces from distribution theory. $\mathcal{S}$ the space of rapidly decreasing smooth functions. $\mathcal{S}'$ the space of tempered distributions, dual ...
2
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0answers
22 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...