Use this tag for questions about distributions (or generalized 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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About test functions for supersolutions

Let $B_{1}$ the unit open ball in $\mathbb{R}^{n}$ and $u \in H^{1}(B_{1})$. For $k,m >0$, let $\bar{u} = u^{+} + k$ and $\bar{u}_{m} = \bar{u}$ if $u < m$ and $\bar{u}_{m} = k+m$ if $u \geq m$. ...
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1answer
23 views

Distributional derivative with discontinuities

Suppose that $f$ is continuously differentiable on $\mathbb{R}$ except at $x_1,\cdots,x_m$ where $f$ has jump discontinuities, and that its pointwise derivative $df/dx$ (defined except at $x_1,\cdots,...
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2answers
29 views

Fundamental Solution to 2nd Order ODE

I'm currently doing a problem with the fundamental solution for $$-u''+k^2u=f(x) \quad , \quad -\infty < x < \infty$$ I'm wondering if fundamental solutions are supposed to satisfy the ...
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1answer
58 views

Support of a Distribution

Let $U$ be a nonempty open subset of $\mathbb{R}^n$ and $\mu$ be a Radon measure on $U$. Define $$T_\mu(\phi) = \int_U \phi d\mu$$ for all $\phi \in D(U) = C_c^\infty(U)$. Prove that $T_\mu$ is a ...
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1answer
63 views

Why are these distributions positive?

I am trying to understand some calculations in a paper by Sidney Coleman. He is showing that certain distributions are positive. The paper can be found here. What I am talking about is happening at ...
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1answer
64 views

Space of test functions defined by norms

This is the problem assigned: So I know that a locally convex Hausdorff space is defined by a vector space and a family of seminorms. So is part $a$ just wanting me to show that $\|\phi\|_m$ is in ...
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1answer
11 views

Prove version of Bernsteins inequality $||\partial^{\alpha}f||_{L^{\infty}}\leq CR^{|\alpha|}||f||_{L^{\infty}}$

This is the question I am trying to answer, I am having difficulties understnading what is going on. My first question is there a typo in the hint, i.e should it be a new function $g=f\ast h_{1/R}$ or ...
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1answer
58 views

Decomposition of complex Radon measures

Suppose you have a complex Radon measure $\mu$, treated as a distribution. Then does every such Radon measure admit a decomposition of the form $\mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f$ ...
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1answer
30 views

Obtain a tempered distribution from $1/|x|$ by subtracting a multiple of $\phi(0)$

I am trying to show that for an appropriate choice of constants $c_{\delta}$ which diverge as $\delta \to 0$ a distribution $W\in \mathcal{S}'(\mathbb{R})$ can be defined by: $$ W(\phi)=\lim_{\delta \...
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19 views

Multiplying the PV$(\frac{1}{x})$ by $x$

I am trying to show that $x\text{PV}\left(\frac{1}{x}\right) = 1$ in the sense of distributions, that is $\langle x\text{PV}\left(\frac{1}{x}\right), \phi \rangle = \langle 1, \phi \rangle$ for all $\...
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23 views

Distributional derivative of Indicator function $\times$ smooth function

I have a question about distributional derivative. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$. Suppose $\Omega_{1} \subset \Omega$ has the following property: $f \in C_{c}^{\infty}(\...
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1answer
15 views

Let $T : \mathcal{D}(\mathbb{R}) \to \mathbb{R}$ be given by $T(\phi) = |\phi(0)|$. Show that $T$ is not a distribution.

As the title states, I wish to show that $T(\phi) = |\phi(0)|$ is not a distribution. I assume I need to show that the bound $|T(\phi)| \leq C \sum_{|\alpha| \leq n} ||D^{\alpha}\phi||_{L^{\infty}}$ ...
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1answer
48 views

Second derivative of the delta function

Is the second derivative of the delta 'function' even? My intuition tells me yes, and my calculation relies on delta''(-x) = delta''(x).
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0answers
45 views

Is there a Plancherel-type identity for generalized Fourier Transforms?

Let $S$ be in $\mathcal{T}$, the set of tempered distributions, and $\mathcal{F}S$ be its Fourier Transform. Is there some relationship for such distributions, analogous to the Plancherel Theorem for $...
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1answer
41 views

Second order differential equation with Heaviside function

I have a differential equation of the form $$y''(x) - a y(x) + b \theta(c - x) = 0, \quad y(0) = 0, \quad \lim_{x \to \infty} y(x) = 0,$$ where $a$, $b$, $c$ are some constants and $\theta(с - x)$ is ...
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1answer
30 views

Let $\phi\in\mathscr{D}$. Then $f\phi\in\mathscr{D}$ for every smooth function $f$.

Let $\phi\in\mathscr{D}$, where $\phi$ is a test function and $\mathscr{D}$ is the set of all test functions. Then $f\phi\in\mathscr{D}$ for every smooth function $f$. This one seems...trivial. So ...
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1answer
23 views

Properties of convergence on the set of test functions

I'm trying to prove the properties of convergence on the set of test functions, $\mathscr{D}$, but the following is giving me some problems. Let $\phi_n\to\phi$ and $\psi_n\to\psi$ on $\mathscr{D}$. ...
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1answer
18 views

Prove that the functional $F$ on $\mathscr{D}$ defined by $\langle F, \phi\rangle=\int_{\mathbb{R}^n} f\phi$ is a distribution.

Let $f$ be a locally integrable function on $\mathbb{R}^n$. Prove that the functional $F$ on $\mathscr{D}$ defined by $\langle F, \phi \rangle = \int_{\mathbb{R}^n} f\phi$ is a distribution, where $\...
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28 views

Show that $f_n\to 0$ in the distributional sense.

Let $f_n(x)=\sin{(nx)}$. Show that $f_n\to 0$ in the distributional sense. I know that this is true only if $\langle f_n,\phi\rangle=\int_{\mathbb{R}^n} f_n\phi\to \int_{\mathbb{R}^n} f\phi=\langle f,...
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53 views

Integrating with a Dirac delta function $\delta(x-a)$ when $a$s not in the domain of integration?

The delta function has the fundamental property that \begin{align} \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) \end{align} and, in fact, \begin{align} \int_{a-\epsilon}^{a+\epsilon}f(x)\delta(x-a)...
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35 views

About Semiclassical Analysis and other

I read something about this theory. I honestly do not care to find out the link between quantum mechanics and general relativity, because it's too much for me. But I have seen that there are still ...
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25 views

Representation of the delta distribution as an element of the dual of $H^1$

I'm working with some Sobolev spaces and I just wanted to consider the elements of $H^{-1}$ as elements on $H^1$ (Riez Theorem). Since the delta function $\delta(f) = f(0)$ is an element of the dual ...
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Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
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15 views

Transport PDEs with mixed linear and nonlinear terms and distribution solutions

This question is concerned with the theory of solutions of first order transport-type PDEs that are linear in some variable and nonlinear in others. E.g. this beauty: $\frac{\partial u}{\partial t}+\...
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2answers
61 views

Show that $f(\phi)=\sum_{k=0}^\infty \phi^{(k)}(k)$ for $\phi \in D$ has no finite order

Let $\phi \in D:=D(K)=C^\infty_0(K) $ be a test function and let $f \in D^*=\{f:D \to \mathbb{R} : f \text{ bounded and linear} \} $ be a distribution. A distribution has finite order if: $$\exists ...
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1answer
32 views

Singular integral operator

i got the following problem to solve. Let $0 < \alpha < 1$, $L \in L_\infty([0,1]^2)$, $D = \{(x,y) \in \mathbb{R}^2: x = y\}$ the diaagonal of $\mathbb{R}^2$ and $k:[0,1]^2 \setminus D \to \...
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1answer
49 views

Rellich's theorem fails at the end point.

If $\{f_k\}\subseteq H^s(\Bbb{R}^d)$ is a bounded sequence and support of each $f_k$ is contained in a common compact set, then there exists a subsequence that converges in $L^q$ for $\forall$ $q, 2\...
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23 views

Bounded sequence in Sobolev space has a convergent subsequence

Suppose $\{\phi_k\}$ are a bounded sequence of distributions on in $H^s(\Bbb{R^d})$, where $0<s<\frac{d}{2}$ and $H^s$ is the Sobolev space. Then the sequence has a subsequence that converges in ...
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1answer
22 views

Definition of homogeneous distribution.

I ran into the following definition: If $u$ is a distribution on $\mathbb{R}^d$, then $u$ is called homogeneous of order $m$ if $u(\lambda x) = \lambda^m u(x)$, $x\in\mathbb{R}^d$. But $u$ is not ...
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Proof that inequality holds

Theorem: Let $u \in D'(\Omega)$ and $K \subset \Omega$, $K$ compact $$\exists \lambda \in \mathbb{N} \text{ and } c \geq 0 \text{ such that } \\ |\langle u, \phi \rangle| \leq c \sum_{|a| \leq \...
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24 views

Convolution of a distribution with a $C^{N}$ function.

I've been working on the following Problem from Friedlander's introduction to the theory of distributions: Show that if $u$ is a distribution of order $N$ and with compact support on $\mathbb{R}^...
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1answer
35 views

Second distributional derivative of cosine

I need to compute second distributional derivative of the function $$ g(x) = cos|x-2|, $$ but I'm not sure about my solution. \begin{align} \left<g'', \varphi \right> = \left<g, \varphi''\...
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1answer
47 views

Convolution of Schwartz and test function approximated by partition of unity.

Let $\rho\in\mathscr{D}$, $0\leq\rho\leq 1, \rho(0) = 1$, and $\sum_{n\in\mathbb{Z}^d}$ $\rho(x-n) = 1$. Denote, $\rho_{n,\epsilon}() = \tau_n\rho(\frac{x}{\epsilon})$, where $\tau$ is the translation ...
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1answer
26 views

$r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding

Show that $r:\mathscr{S'}\rightarrow\mathscr{D'}$, $u\mapsto u|_{\mathscr{D}}$ is not a topological embedding. For this problem, would it suffice to construct a sequence $\{u_n\}$ in $\mathscr{D'}$ ...
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1answer
27 views

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$. I am not quite sure how to start this problem. ...
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69 views

Prove that $e^x$ is not a tempered distribution on $\mathbb{R}$

Consider the following sequence of functions $\psi_n(x) = e^{-(1+\varepsilon)x} \dfrac{1_{|x|\leq n}}{n}$. Clearly, $|\psi_n^{(m)}(x)|\leq\dfrac{(1+\varepsilon)^m}{n}$. Hence, the $\psi_n$-s are ...
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1answer
93 views

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$?

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$? Where $\mathcal{D}(\Omega)$ is the space of test functions with support compact and $\mathcal{D}'(\Omega)$ is the ...
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1answer
55 views

Distributional derivatives

I need to compute derivatives as distributions of following functions: $f(x) =$ $|x|$ $|x^2 - 1|$ $\mathrm{sgn}(x)$ $4$ Where $f : \mathbb{R} \to \mathbb{R}$. ad 1) $|x|$ is continuous, so it ...
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1answer
67 views

Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...
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42 views

Sobolev and Fourier

If we have $f=1_{[a,b]} \varphi$ with $\varphi \in \mathcal{D}(\mathbb{R})$, we found that the sufficient and necessary conditions to have $f\in H^1$ is that $\varphi(a)= \varphi(b)=0$. If we take $\...
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76 views

How does the Dirac delta function operate when its peak is at the boundary of an integral?

As far as I can tell the Dirac delta function in an integral picks the value of the multiplying function at the peak provided the peak is within the boundary, i.e. $$\int^{a+e}_{a-e} \delta (x-a) f(x)...
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32 views

Convolution of Schwartz Function and Distribution of Compact Support

From Stein-Shakarchi Functional Analysis Chapter 3 Exercise 12 and Exercise 13. I'm having trouble proving that: If $F_1$ is a distribution with compact support and $\varphi\in \mathcal{S}$ is a ...
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35 views

Convolution of two distributions

Consider the convolution product: $$H(x)\ast\operatorname{Pf}\dfrac{H(x)}{x},$$ where $\operatorname{Pf}$ denotes pseudo function. This means, that $\operatorname{Pf}\dfrac{H(x)}{x}$ is, as defined ...
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1answer
44 views

Do tempered distributions form a topological subspace of the space of distributions?

I'm learning about distributions and tempered distributions. From what I understand, by "enlarging" the space of test functions $\mathcal{D}$ to the Schwarz space $\mathcal{S}$ and correspondingly "...
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0answers
11 views

$\Lambda\psi$=0 if $(D^{\alpha}\psi)(x)$ for every $x\in$ supp $\Lambda$ and every multi index $\alpha$

I can show that $\Lambda\psi$=0 if $(D^{\alpha}\psi)(x)$ for every $x\in$ supp $\Lambda$ and every multi index $\alpha$ given the support of $\Lambda$ is compact. But how one extend this argument for ...
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2answers
49 views

Continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is a distribution.

Prove that every continuous linear functional on $C^{\infty}_{loc}(\mathbb{R}^d)$ is of the form $\Lambda\mapsto\Lambda f$ for some distribution $\Lambda$ with compact support. I am stuck at this ...
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39 views

Positive distribution $\Lambda$ as positive Radon measure

Exercise 4 of Chapter 6 in Rudin's Functional Analysis states that every "positive" distribution $\Lambda\in D^{'}(\Omega)$, i.e, $\Lambda\psi\geq 0$ whenever $\psi\in D(\Omega)$, is a positive ...
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29 views

Multi-index notation and differentation

For example, let $\Omega \subseteq \mathbb{R}^n$ open, and $C^\infty(\Omega):=\lbrace f: \Omega \longrightarrow \mathbb{C} : f$ $\mathrm{regular}\rbrace$. For $\alpha = (\alpha_1,...,\alpha_n) \in \...
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1answer
37 views

Convergence in distributions

Let $\varphi ∈D$ be a test function on $\Bbb{R}$. Is the sequence $f_n(x)=\frac{\varphi(nx)}{n}$ convergent in the test function space $D$? What is the limit? Please provide a hint to start.
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1answer
57 views

convergence of sequence in a Distribution

Let $\varphi \in D$ be a test function on $\Bbb{R}$. Is the sequence $f_n(x)=\frac{1}{n}\varphi(\frac{x}{n})$ convergent in the test function space $D$? What is the limit? please a hint to start.