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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Definition of the convolution with tempered distributions and Schwartz function

If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$, where we place ...
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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
31 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, ...
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20 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
17 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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28 views

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 ...
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1answer
34 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, ...
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1answer
46 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
26 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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63 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
81 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
52 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 ...
2
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1answer
62 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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0answers
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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67 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) ...
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26 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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0answers
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
42 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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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
47 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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0answers
36 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.
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differential of the regular distribution in the space $D'$

Determine the differential of the regular distribution $T_f$ in the space $D'$(continuous dual of $D$) for $f(x)=H(x)cos(x)$, where $H$ is a Heaviside function and $x\in \Bbb{R}$. Since $H(x) = +1$ ...
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1answer
36 views

How to compute the fourier transform of $\operatorname{sgn}$ directly?

I've been trying to compute the fourier transform of $\operatorname{sgn}(x)$, but I'm having trouble with the complex exponential at infinity. The issue is the following: by definition we have ...
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84 views

How do I show this is a solution for this differential equation?

Consider that $$E(t,x)=\dfrac{H(t)}{2\sqrt{\pi t}}e^{-|x|^2/4t}.$$ I want to show that $E_t - E_{xx} = \delta(t)\delta(x)$. This means that we need to show that if $\phi\in ...
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2answers
83 views

Derivative of $\ln |x|$ in the distributional sense

Consider the function $\ln |x|$, since it is locally integrable we can form the distribution $$(\ln |x|,\phi)=\int_{-\infty}^{\infty}\ln |x|\phi(x)dx.$$ Now, I want to show that in the sense of ...
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54 views

How to define composition of distribution with a function correctly?

Recently I've been reading some notes on distribution theory and the author makes the following definition: Let $\zeta\in \mathcal{D}'(\mathbb{R})$ be a distribution and $f$ a $C^\infty$ function, ...
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1answer
27 views

Differential operator- Equality

Suppose that $L$ is a linear differential operator such that $Lu(x)=f(x)$. Why does the following equality hold? $$L \frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}^n} \hat{y}(\omega) e^{i \omega t} d ...
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Differential operator and multi-index

By induction it can prove Leibnitz rules $\displaystyle D^\alpha(fg)=\sum_{|\beta| \leq |\alpha|} \binom{\alpha}{\beta} D^\beta f D^{\alpha - \beta} g$ from the book where I'm studying, it says that ...
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1answer
39 views

Question about Distributional Derivative of Monotonically Non-decreasing function

Suppose $f \geq 0$ on $\mathbb{R}$ be monotonically non-decreasing. Let $T_f$ be the distribution given by $f$. Then, $T_f \geq 0$ for all $\psi \in C^{\infty}_c(\mathbb{R})$ with $\psi(x) \geq 0$ on ...
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38 views

Is the computation of this limit of distributions done right?

I've been trying to show that $$\lim_{t\to \infty} \dfrac{e^{ixt}}{x-i0}=2\pi i \delta(x).$$ For that I've used the fact that $$\dfrac{1}{x-i0}=\lim_{\epsilon\to 0^+} \dfrac{1}{x-i\epsilon}=i\pi ...
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1answer
36 views

Limit of distributions

I'm trying to solve the following excercise, whitout any luck. Let $\eta(x)=\begin{cases} c \exp\left(\dfrac{1}{|x|^2-1}\right), & \text{if} \;|x| \leq 1 \\ 0 & \end{cases} $ $\qquad ...
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1answer
46 views

How to show this sequence is a delta sequence?

Consider the sequence $(\phi_n)_{n\in \mathbb{N}}$ of test-functions $\phi_n\in \mathcal{D}(\mathbb{R})$ defined by $$\phi_n(x) = \dfrac{n}{\sqrt{\pi}}e^{-n^2x^2}.$$ I want to show that this is a ...
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56 views

How to show this property of the delta function?

Let $\mathcal{D}(\mathbb{R})$ be the space of test-functions in $\mathbb{R}$ and let $f$ be a $C^\infty$ function. I want to show that if $f$ has $n$ zeroes $x_1,\dots,x_n$ in the interval where it is ...
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21 views

Fourier transform defines a tempered distribution

I have taken the Fourier transform of the wave equation in $\mathbb{R}^3$ and solved to get $\tilde{u}(\xi,t) = \tilde{f}(\xi) \cos( |\xi|t )+\tilde{g}(\xi) \sin( |\xi| t)/|\xi|$ $f,g$ are in ...
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1answer
32 views

Prove that $\rho_n \star f \to f$ in $L^p(R^N)$.

Let $\rho \in L^1(R^N)$ with $\int_{}^{} \rho=1$ .Set $\rho_n(x)=n^N\rho(nx)$. Let $f\in L^p(R^N)$ with $1\leq p<\infty$. Prove that $\rho_n \star f \to f$ in $L^p(R^N)$. My try: Since $f \in ...
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1answer
34 views

Dirac delta and test functions

By the use of the test functions prove that: $$f(t) \delta '(t) = f(0) \delta '(t) - f'(0) \delta(t)$$ Attempt: I want to use some test function $\phi (t)$. So starting from $$\langle f \delta' , ...
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27 views

Translation-invariant distribution

This question is exercise 6.27 in adult Rudin. The problem: Find all distributions $u \in \mathscr{D}(\mathbb{R}^n)$ that satisfy at least one of the following two conditions: (a)$\tau_xu=u,\ ...
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80 views

An exercise about distribution in Rudin

This exercise is 6.19 in Rudin's Functional analysis. The Problem: $\Lambda \in \mathscr{D}'(\Omega), \ \phi \in \mathscr{D}(\Omega), \ (D^{\alpha}\phi)(x)=0, \ \forall \ x$ in the support of ...
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18 views

Fourier transform of $\|x\|^{-n}$ in $\mathbb R^n$ with cut-off at $0$

I have read in Grafakos, "Classical Fourier analysis", Example 2.4.9, that the Fourier transform of $\eta(x)\|x\|^{-\alpha}$ in $\mathbb R^n$, $0<\mathrm{Re}(\alpha)<n$, is explicitly known, ...
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39 views

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

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |(1+|x|)^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ ...
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0answers
19 views

Test function for proof of moser harnack inequality

There exists a standard method to build test functions? I would like to construct a sequence of functions $\phi_{m} \in C^{\infty}_{c}(B_{1})$ such that $\phi_{m} \to \psi{u}^{-2}$, where $\psi \in ...
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1answer
29 views

Is there is notion of Fourier transform of distribution?

We note that every tempered distribution is a distribution. Can we find a example of distribution which is NOT a tempered distribution? Can we talk of Fourier transform of that ...
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0answers
26 views

Convergence in convolution

This is an exercise in Rudin's Functional analysis, which is 6.23. The problem: $f_i \in L_{loc}^1(\mathbb{R}^n), \ \lim\limits_{i \to \infty}(f_i*\phi)(x)$exists, $\forall \phi \in \mathscr{D}, \ x ...
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0answers
56 views

How to show this limit of distributions?

I'm trying to solve the following exercise regarding limits of distributions: Establish the following limit (on the distributional sense) $$\lim_{t\to 0\pm}\ln (\tau + it) = \ln |\tau| + i\pi ...