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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2
votes
1answer
43 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
votes
1answer
30 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
votes
2answers
34 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 ...
0
votes
1answer
15 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. $P \delta(x) \in C^{\infty}(\mathcal{R})$?
0
votes
2answers
45 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
65 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
votes
1answer
40 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 ...
1
vote
1answer
30 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
votes
1answer
34 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 $\int_C u(x,y)\ dC$ ...
2
votes
0answers
50 views

Show that $e^{-a|x|}$ does not belong to Schwartz space

Let $f : \mathbb R \to \mathbb R$ and $a > 0$ given by $f(x) = e^{-a|x|}$. Show that $f$ is rapidly decreasing and belongs to $L_1(\mathbb R)$, but not to $\mathcal S(\mathbb R)$. I had shown that ...
1
vote
1answer
52 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
votes
0answers
28 views

Solution of nonlinear Schrödinger equation

Consider the linear Shr\"odinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
0
votes
1answer
31 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: ...
1
vote
2answers
47 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 $T$ ...
0
votes
1answer
12 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 ...
1
vote
0answers
70 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
0answers
29 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 ...
1
vote
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"). ...
1
vote
1answer
38 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
vote
1answer
18 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 ...
0
votes
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
votes
0answers
61 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 ...
0
votes
0answers
42 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
votes
1answer
42 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
votes
1answer
44 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
49 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
votes
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 ...
0
votes
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 ...
1
vote
0answers
44 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 ...
1
vote
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 ...
1
vote
1answer
54 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) = ...
1
vote
0answers
32 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
votes
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 ...
0
votes
1answer
56 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 ...
-2
votes
1answer
57 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 ...
0
votes
0answers
31 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
votes
0answers
39 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
votes
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 ...
0
votes
0answers
45 views

Question with tried Eigenvalues of Laplacian operator and Sobolev spaces III.

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 ...
0
votes
0answers
20 views

eignvalues of laplacian operator and Sobolev spaces -II

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, Let $F=(F_t) \in C^0(I,L^2(\Omega))$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od the ...
1
vote
1answer
28 views

Why do test functions have to be real valued?

Seems like if functions can be made to be complex-valued, they're going to be complex-valued. I have not seen anything about why test functions must be real valued.
1
vote
2answers
192 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
0
votes
0answers
30 views

Eigenvectors of Laplace operator on a particular functiona space

Exercise 8.15 in [1] is: "Show that $\Delta u=\lambda u$ has no solutions of polynomial growth if $\lambda > 0$, but does have such solutions if $\lambda < 0$." How should I make sense of this? ...
0
votes
0answers
47 views

Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
1
vote
1answer
48 views

show a PDE has no distribution solution in $\mathbb{R^2}$\{0}

" Consider the following equation in the plane $x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=f(x^2+y^2)$ where $f(t)$ is a $C^\infty$ function of the real variable $t$ such that ...
1
vote
1answer
56 views

Fourier transform of $\frac{x_i^2}{|x|^2}$

For a function $f$ in $L^1(\mathbb{R}^n)$, it is natural to define the Fourier transform as $$\mathscr{F}(f)(\xi)=\int_{\mathbb{R}^n}f(x)e^{-ix\cdot \xi}dx.$$ And the we may extend it to rapidly ...
1
vote
0answers
36 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
0
votes
1answer
63 views

Verify that a function is a solution to the 3-dimensional wave equation.

For $n=3$. How to verify that a smooth function $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{*} $$ is a solution to the 3-dimensional wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad ...
0
votes
0answers
111 views

Delta function?

I came across this integral: \begin{align} \int_{-\infty}^{\infty} \mathrm{d}k_1 |f(k_1)| \int_{-\infty}^{\infty} \mathrm{d} k_2 |g(k_2)|& \nonumber \times\bigg\{\lim _{V \to ...