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).

learn more… | top users | synonyms

0
votes
0answers
57 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 ...
1
vote
0answers
40 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 ...
0
votes
0answers
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) ...
2
votes
2answers
102 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}, ...
1
vote
0answers
21 views

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 ...
1
vote
2answers
70 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 ...
-1
votes
2answers
63 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 ...
0
votes
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 ...
2
votes
1answer
134 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 ...
3
votes
1answer
100 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 ...
4
votes
0answers
55 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
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 $$ ...
0
votes
0answers
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
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} ...
3
votes
0answers
39 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 ...
1
vote
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 ...
7
votes
1answer
278 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 ...
2
votes
2answers
67 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
votes
0answers
35 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 ...
3
votes
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) = ...
6
votes
1answer
70 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 ...
0
votes
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.
0
votes
0answers
18 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 ...
0
votes
1answer
42 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
votes
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= ...
1
vote
2answers
28 views

Green and heaviside function

I have a question concerning the Wikipedia's article on green function stating in the section "table of green fuctions", that a green function satisfying: $$(\gamma+\partial_t)G(t)=\delta(t)$$ has an ...
0
votes
1answer
35 views

Derivative as distribution

We consider the Heaviside function $H(x)$. $H'(0)$ doesn't exist. The derivative exists if we define $H$ as a distribution $$H: \phi \to \int_{-\infty}^{+\infty} H(x) \phi(x) dx= \int_0^{+\infty} ...
8
votes
2answers
96 views

Can a “continuous” convex combination not be element of the convex hull?

Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ...
1
vote
2answers
65 views

Is the function $|x|$ in $W^{1,p}$?

I have the following question: We consider in the segment $I=]-1,1[$, the function $f(x)=|x|.$ The question is: For each value $p \in [1,+\infty[$ do we have $f \in W^{1,p}(I)$? My purpose is: We ...
0
votes
0answers
15 views

Understanding an example of Renardy's book (example 5.48)

In the example $5.48$, authors say that: To prove that a sequence of integrable functions $f_n: \mathbb{R} \to \mathbb{R}$ converges to the delta function, it suffices to show that the primitives ...
0
votes
1answer
26 views

Show that $u_{tt}=u_{xx}$ in the sense of Distributions

Let $u(x,t)=f(x+t)$ , where $f$ is any locally integrable function on $\mathbb{R}$. Show that $u_{tt}=u_{xx}$ in the sense of Distributions My try: For $\phi \in D(\mathbb{R})$, ...
0
votes
1answer
24 views

Describe the distributional derivative of $f$

Let $f$ be a piece wise defined function with piece wise continuous derivative. Describe the distributional derivative of $f$. My try: If I suppose that the jump discontinuities are at the points ...
1
vote
0answers
18 views

Distribution functions: differentials in the numerator or denominator

One paper I'm looking at says, $n(M, z) \, dM \, dz$, the number of sources with mass $M$ at a redshift $z$, in the mass interval $dM$ occurring in the redshift interval $dz$. While another ...
1
vote
1answer
44 views

Example of a linear functional, but not a distribution

I'm looking for an example of a linear functional $u: C_c^\infty(\Omega) \to \mathbb C$ ($\Omega \subset \mathbb R^n$ open), which is not a distribution. I could not find anything... I thought of ...
0
votes
0answers
20 views

Distributions on compact and semi-open intervals

In the theory of distributions (aka generalized functions), one considers mostly distributions $T \in \mathcal{D}(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$. Hereby, the space ...
2
votes
1answer
79 views

Derivative of Dirac delta function as a measure

Dirac delta function can be defined in several ways. I know two definitions. One is as a distribution and the other is as a measure. I found many materials on the derivatives of delta function as a ...
2
votes
0answers
39 views

Uniform convergence of compactly supported function

If $f\in C_c^\infty$, i.e. Compactly supported, then for $g_k(x)=f(kx)$, would $g_k\to 0$ uniformly as $k\to \infty$ since g vanishes outside a given compact set which won't be touched for $k$ ...
0
votes
1answer
44 views

existence of smooth functions with support $0$

In my real analysis homework, theres a question which begins with "let $F$ be a distribution on $\mathbb{R}^n$ such that $\operatorname{supp}(F)=\{0\}$" This means that the only $\phi \in ...
0
votes
0answers
139 views

Folland real analysis 9.11

This comes from question 9.11 of Folland's Real analysis textbook. Unfortunately, I have no idea to how to start with this question. So can some one help me with part $a$? For part $a$, I can not ...
1
vote
0answers
20 views

Behaviour of functions in weighted sobolev spaces

If $f$ and $Df$ are in $L^2(\mathbb{R}, e^{u^2} dx)$, can we say $f(u)e^{\frac{u^2}{2}}$ is bounded. Here $Df$ distributional derivatie of $f$. That is, If $\int_{\mathbb{R}} \lvert f(u) \rvert^2 ...
1
vote
1answer
91 views

Proof of regular version of the Urysohn lemma

I know it's a well-known result, but I have not found any clear formalization, and I need a clear formalization. So I want to know if you agree with this formalization, and this proof. Thank you for ...
1
vote
0answers
54 views

topology of $C^\infty _K(\Omega)$

Let be $\Omega \subset \mathbb R^n$ and open set. The space of smooth function $C^\infty (\Omega)$ is endowed with the topology generated by the seminorms $\{p_{K,m}: \;K \subset \Omega ...
0
votes
0answers
20 views

Is this distribution regular?

I have to check whether the following distribution is regular: a) $$ [|3x-1||_{[0, \infty]}]'' $$ b) $$ [|3x-1||_{[-\infty,0]}]'' $$ I've got that: a) $$[|3x-1||_{[0, \infty]}]''(\Phi) = ...
1
vote
0answers
32 views

Prove that Heaviside's function H(x) is a solution for equation

Prove that Heaviside's function H(x) is a solution for equation $$ \frac{\partial^2 u}{\partial x_1\partial x_2}=0 $$ $$$$ $$$$ For example, to solve that function $$ F(x)=e^{2x}H(x)$$ is solution ...
4
votes
1answer
97 views

A commutation between curl and integral

I have been struggling to understand the only derivation of Ampère's law from the Biot-Savart law for a tridimensional distribution of current (which, needless to say, is not the case of a linear ...
0
votes
0answers
40 views

Topology for distributions on a compact space

I'm having trouble in distribution theory, though not in the usual setting. The context is in theoretical physics, trying to solve BF theory. My goal is to solve the following equation: $$\forall i ...
0
votes
1answer
23 views

Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

For a fixed $y$ $$\phi_{y,\epsilon}(x)= \left\{ \begin{array}{ll} \exp(-\frac{\epsilon^2}{\epsilon^2-|x-y|^2}) & \mbox{if $|x-y| \lt \epsilon$};\\ 0 & ...