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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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 $H(x) = +1$ ...
2
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1answer
37 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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0answers
85 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 $\phi\in \mathcal{D}(\mathbb{R}^2)$...
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2answers
86 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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0answers
57 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
28 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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26 views

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 ...
2
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0answers
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
38 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
47 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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0answers
63 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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0answers
22 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 L^...
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1answer
37 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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0answers
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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0answers
81 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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0answers
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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2answers
41 views

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

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |(1+|x|)^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ $\|f\|_{(\...
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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 H^...
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1answer
30 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 distribution?...
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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
58 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 H(-...
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0answers
42 views

Is this how we define “limit in the distributional sense”?

Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over $\mathcal{D}(\mathbb{...
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24 views

Property of differential operator

We have $P(x, \partial)=\sum_{\alpha \in A} a_{\alpha}(x) \partial^{\alpha}$ where $A$ is a set of multiindices Definition: A distribution u is called fundamental solution if $P(x, \partial) F=\...
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2answers
103 views

Is Plancherel's theorem true for tempered distribution?

Let $f, g\in L^{2},$ by Plancherel's theorem, we have $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle.$$ My Question is: Is it true that: $$\langle f, g \rangle= \langle \hat{f}, \...
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Hadamard finite part and pricipal part of Laurent series

In "Fourier analysis and its applications" by Folland, the distribution (generalized function) $X^{-k}$ is defined as $$ X^{-k}[\phi] = \frac{1}{(k-1)!} P.V. \int \frac{\phi^{(k-1)}(x)-\phi^{(k-1)}(0)}...
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2answers
74 views

Delta distribution doesn't belong to any $L^p$ but there is a $H^1_0$ representation of it, why?

Let $\Omega := (-1,1)$. I heard that the delta distribution $$\delta\colon H^1_0 \to \mathbb{C},\, \phi \mapsto \phi(0), \,\, \delta \in H^{-1} := (H^1_0(\Omega))'$$ has a Riesz representation in $H^...
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2answers
66 views

Proof of uniqueness about distribution in Rudin's

I'm reading Functional Analysis by Rudin, and have trouble understanding a part of the proof of theorem 6.33, in page 174. This theorem states an one-to-one relationship between a linear continuous ...
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1answer
80 views

Distributional solution of the heat equation

I want to show that $\frac{\partial{E}}{\partial{t}}-\Delta E=\delta(t,x) $. So it suffices to show that $\langle \frac{\partial{E}}{\partial{t}}-\Delta E, \phi \rangle=\phi(0,0) $. So far I have ...
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1answer
153 views

Tempered distributions and convolution

I remember that if $f,g \in \mathcal{S}(\mathbb{R}^n)$ , then it is well-defined \begin{align*} \displaystyle (f \ast g)(x)= \int_{\mathbb{R}^n} g(x-y)f(y)dy=\int_{\mathbb{R}^n} (\tau_x \widetilde{g})...
3
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1answer
105 views

Solve the differential equation $x^2u'=0$ in the sense of distributions

Solve the differential equation in the sense of distribution: $$x^{2}\frac{du}{dx}=0$$ This is from "Principles of Applied Mathematics" by Keener, problem 4.1.5. The solution in the back of the ...
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0answers
58 views

The space of test-functions carries any other structure on it?

I'm starting to study distributions and on the lecture notes I'm reading the author defines a test-function as a function $f : U\subset \mathbb{R}^n\to \mathbb{R}$ which is infinitely differentiable ...
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0answers
34 views

The derivative of the Dirac delta as a distributional limit in the form of an improper integral?

Let $f(x, y)$ be a function. I want to define a distribution with it. If $$\lim_{a\to\infty} \left< \int_{-a}^a f(x, y) dy, \phi(x) \right> = \lim_{a\to\infty} \int_{-\infty}^\infty \int_{-a}^a ...
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29 views

Is convolution of a distribution with a function defined?

I know how the convolution of a distribution is defined. But when my teacher defined the Hilbert transform of a $f\in\mathcal{C}^1(\Bbb R)\cap L^1(\Bbb R)$ as $$ Hf(x):=\frac1{\pi}\operatorname{p.v.}\...
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25 views

Calculate $L^1$ norm of second derivative of a n-periodic sawtooth function constructed with a step function

can someone help me calculating the $L^1$ norm of the second derivative of a n-periodic sawtooth function defined as follows: $$ f_n(x) \doteq \frac{u(nx)}{n}-kx $$ where $k\in(1,2)$ and all I know ...
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0answers
34 views

Fourier transformation of piece-wise function

Let the function $ f(x)= \begin{cases} 0: &|x|>1\\ 1: &|x| \leq 1 \end{cases} $ $|x|$ is the euclidian norm of $x$. My question is how we calculate $F(f)$? (the Fourier transformed). I ...
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1answer
84 views

Inequality about disbtribution in Functional Analysis by Rudin

I'm reading Functional Analysis by Rudin about distribution theory. I have a problem of derivation of the inequality (10) in theorem 6.25, page 166. It first proves an inequality $$(5)\quad\quad\quad\...
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1answer
18 views

Principal value of $1/x$ does not arise from either a locally integrable function or a Radon measure

The distribution $\text{p.v.}1/x\in C_c^\infty(\mathbb{R})^*$ is defined by the formula: $$\left\langle f,\text{p.v.}\frac{1}{x}\right\rangle:=\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon} \frac{...
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0answers
40 views

Show that a regularizing operator $K : C_c(\Omega) \to \mathcal{D}'(\Omega)$ has kernel $k \in C^\infty(\Omega \times \Omega)$.

I am reading Francois Treves' Introduction to pseudodifferential and Fourier integral operators, vol. I. Let $\Omega \subseteq \mathbb{R}^n$ be open. On page 11, Treves defines what it means for a ...
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1answer
48 views

$W^{1,\infty}(\mathbb{R})$ is the same as $C^{0,1}(\mathbb{R})$

Let $f\in (C_c(\mathbb{R}))^*$ be a distribution. Show that $f\in C^{0,1}(\mathbb{R})$ if and only if $f\in L^\infty(\mathbb{R})$, and the distributional derivative $f'$ of $f$ also lies in $L^\infty(\...
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1answer
281 views

Why is multiplication on the space of smooth functions with compact support continuous?

I was reading Terence Tao post https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/ and i'm not able to prove the last item of exercise 4. I have a map $F:C_c^{\infty}(\mathbb R^d)\...
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2answers
98 views

Show that the only tempered distributions which are harmonic are the the harmonic polynomials

Let $d\geq 1$. Using the Fourier transform, show that the only tempered distribution $\lambda \in\mathcal{S}(\mathbb{R}^d)^*$ which are harmonic (by which we mean that $\Delta \lambda=0$ in the sense ...
2
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0answers
38 views

Hahn Banach Theorem extending distribution

For any given distribution $T\in D'(\Omega)$, could $T$ has a coutinuous extension $$\widetilde{T}:C_0(\Omega)\rightarrow R,\ \ \ \widetilde{T}\in(C_0(\Omega))'\ \ ?$$ Could you state a general ...
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5answers
87 views

What are some functions that respect the following criteria? : $f(1/x) = f(x)$ and $\int_{0}^{+\infty} f(x) dx = 1$

I'm looking for some functions that respect these six criteria: $f$ is defined on $[0 ; +\infty[$ $f$ is differentiable everywhere in $[0 ; +\infty[$ $f(0) = 0$ $\lim\limits_{x \to +\infty} f(x) = 0$...
6
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1answer
79 views

Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form (...
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3answers
47 views

How to calculate the distributionderivatives of abs(x)?

Lets say we have $f(x) = |x|$. I want to calculate $f'$ and $f''$, how would I go about this? I understand that this is not defined at $x = 0$, so it will have to be done in two steps.
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0answers
19 views

Every non-negative distribution arises from a non-negative Radon measure

A distribution on $\mathbb{R}^d$ is a continuous linear functional $\lambda: f\mapsto \langle f,\lambda\rangle$ from $C^\infty_c(\mathbb{R}^d)$ (with good seminorms topology) to $\mathbb{C}$. A ...
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1answer
43 views

Does distribution derivative generated by $C^{\infty}$ function forces the distribution to be compactly supported?

Let $\phi \in C_C^{\infty}(\Bbb R)$. Then there exists a $\psi \in C_C^{\infty}(\Bbb R)$ such that $\psi' = \phi \iff \int_\Bbb R \phi = 0$. This is quite easy to be proved. From this it follows that ...
0
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1answer
47 views

How can we apply the formula?

I want to calculate $\Delta \ln{||x||}$ for $x=(x_1, x_2)$. $$\langle \Delta \ln{||x||}, \phi \rangle= \langle \ln{||x||}, \Delta{\phi}\rangle=\int_{\mathbb{R}^2} \ln{||x||} \Delta{\phi(x)} dx= \lim_{...