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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4
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1answer
61 views

Specific problem on distribution theory.

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...
0
votes
0answers
37 views

Dirac Delta Distribution and non-compactly supported test function

I would like to know if there is any problem with defining the following expression: $$ I = \int_0^\infty g(t) \delta(f(t))\mathrm{d}t $$ where $0<\lim\limits_{t\to\infty} g(t) =L<\infty$ and ...
3
votes
0answers
47 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an identity for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group structures ...
3
votes
1answer
46 views

Lebesgue integral of Dirac delta

If I recall correctly, for a bounded function $f$ $$ \int_{\mathbb{R}} f \, d\mu = \int_{\mathbb{R} \setminus \{ a \} } f \, d\mu + f(a) \mu (a).$$ For the Lebesgue measure, $\mu(a) = 0$ and $$ ...
1
vote
1answer
37 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
2
votes
0answers
45 views

What are some properties of the sheaf of distributions?

In a course on measure theory, the lecturer proved that distributions (on a locally convex space I think) form a sheaf $\mathcal D$. He isn't interested in sheaves, so he didn't elaborate. Afterwards, ...
3
votes
1answer
494 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
2
votes
1answer
49 views

Confusion about inclusions of dual spaces

We have the triple inclusions $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$, where the second inclusion is not literal but in the sense of distributions. Related to this answer, why is the ...
0
votes
1answer
42 views

different generalized functions?

I am trying to solve a PDE that's order 1 in time $t\ge0$ and order 2 in space $x\ge0$. The solution $u(x,t)$ exists, is unique and possesses the following properties: $u(x,t)\ge0$ for all ...
2
votes
1answer
46 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
8
votes
4answers
7k views

what is product of delta function with itself?

what is the product of delta function with itself ? what is the dot product with itself ?
3
votes
0answers
98 views

No consistent theory can define a product of distributions: why?

I have been told there cannot be a consistent theory defining a distribution product. Googling for information, I found 1 and 2. Number 1 gives interesting hints on what might happen, and defines a ...
2
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0answers
32 views

Schwartz impossibility result

I was wondering what made it impossible to define a product of distributions. Googling, I found two questions, one of which stated the following impossibility result: There is no associative ...
2
votes
1answer
41 views

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in ...
0
votes
1answer
70 views

Is this operator a Fourier multiplier operator?

I want to study the Fourier transform of $$L_{\alpha}(t) = \frac{e^{i\alpha t}}{t^2} - i\frac{\alpha}{t}$$ Basically i am trying to get a grip on, given a $f$, what is $f(t)\ast L_{\alpha}(t)$ and am ...
5
votes
1answer
59 views

Fourier transform of the Heaviside function

As you can see from the title I want to calculate the Fourier transform of the Heaviside function $u(t)$. Proven the the Heaviside function is a tempered distribution I must evaluate: $$ \langle ...
3
votes
3answers
55 views

Definition in Lax “sequence of continuous functions tending to $\delta$”, are distributions needed for understanding?

I'm trying to read Lax's functional analysis. In chapter 11 he makes a definition which I don't like. A sequence of continuous functions ${k_n}$ on a $[-1,1]$ tends to $\delta$ if $\int_{-1}^{1} ...
0
votes
1answer
56 views

How to make a change of variable inside the Dirac delta?

Let $\delta(\phi) = \phi(0)$ be the dirac delta. I would like to compute $\int_{\mathbb{R}} h(x) \delta(\lambda x) dx$ 1) Since $\delta$ is an unit mass on $0$ $$\int_{\mathbb{R}} h(x) ...
8
votes
1answer
165 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
2
votes
0answers
25 views

Convolution of tempered distributions where one has compact support.

For $u\in\mathcal E'(\mathbb R^n)$ and $v\in\mathcal S'(\mathbb R^n)$, we defined $u\ast v$ by $\langle u\ast v, \phi\rangle = \langle v, \check u \ast \phi \rangle$ for all $\phi\in\mathcal S(\mathbb ...
1
vote
1answer
68 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
8
votes
1answer
115 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
2
votes
3answers
3k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
6
votes
2answers
1k views

Laplacians and Dirac delta functions

It is often quoted in physics textbooks for finding the electric potential using Green's function that $$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$ or more generally $$\nabla ...
2
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0answers
46 views

Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
2
votes
0answers
33 views

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
0
votes
1answer
22 views

Functions and distribution integrals

Suppose $ f, g $ are two smooth functions and that for all $ h: \mathbb{R} \rightarrow \mathbb{R} $: \begin{align*} \int f(h(x)) + g(h(x)) \frac{d^2 h}{dx^2} dx = 0 \end{align*} Can I conclude that $ ...
2
votes
1answer
33 views

ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions

I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be ...
2
votes
0answers
33 views

Check smoothness at point looking at Fourier transform

Let $u \in \mathscr E'(\mathbb R^n)$ we a distribution with compact support. Then $u \in C^\infty(\mathbb R^n)$ if and only if for any $N \in \mathbb N$ there exists $C_N > 0$ such that $$ ...
0
votes
1answer
55 views

Distributions corresponding to $\frac{1}{|x|}$

Stirchartz's book ("A guide to distribution theory and fourier transforms" ) has Chapter 1 exercises Here $\mathcal{D(\mathbb{R}^1)}$ is a set of test functions $\phi:\mathbb{R} \rightarrow ...
9
votes
4answers
277 views

Iterated Limits Schizophrenia

Consider the functions $g_n(x)$, with $n\in\mathbb{N}$, $n \ge 1$ and $x\in\mathbb{R}$, defined as follows: $$ g_n(x) = \begin{cases} 2n^2x & \text{if }0 \le x < 1/(2n) \\ ...
2
votes
2answers
363 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
2
votes
1answer
66 views

Multiplication of a distribution by a continuous function

In my work, I am encountering the issue of having to multiply a continuous function (not necessarily differentiable) by a distribution. It seems to me that if $f(x)$ is a continuous function on ...
1
vote
1answer
27 views

Fourier transform of the principal value distribution

I would like to compute the Fourier transform of the principal value distribution. Let $H$ denote the Heaviside function. Begin with the fact that $$2\widehat{H} =\delta(x) - \frac{i}{\pi} ...
2
votes
1answer
29 views

Weak limits and computing the Fourier transform of the Heaviside function

A common problem on this site is to compute the Fourier transform of the Heaviside function that is $0$ on the negative reals and $1$ on the positive reals. A standard technique is to consider the ...
1
vote
0answers
15 views

Distribution which is not a finite sum of derivatives of continuous functions

Recall that fact that a distribution which is compactly supported can be written as a finite sum of derivatives of continuous functions. This fails for general distributions in $D^\prime ...
0
votes
1answer
23 views

Relation between a distribution in $\mathbb{R}^{1}$ and $\mathbb{R}^{2}$

From Chapter 1, Exercise 8 and 9, Strichartz's book: Distribution theory & fourier transforms Suppose $f$ is a distribution on $\mathbb{R}^1$. Show that $<F,\phi> = ...
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vote
0answers
44 views

Solving Linear ODEs over the space of distributions

I am encountering in my work linear differential equations with coefficients that involve things like Heaviside functions and delta Dirac functions. I know how to find things like Green's functions or ...
0
votes
1answer
34 views

1D green function

I want to solve the problem $$ \frac {\partial^2u}{\partial t^2} + \alpha \frac {\partial u} {\partial t} = f(t)$$ $$u(0)=u'(0)=0$$ $$f(t)=exp(-\beta t)$$ Using distributions (Green functions). ...
0
votes
1answer
21 views

If two distributions vanish on the same set of test functions, then one is a constant multiple of the other

From Chapter 1, Exercise 14, Strichartz's book: Distribution theory & fourier transforms Suppose $f$ & $g$ are distributions such that $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi ...
2
votes
1answer
390 views

Identify distribution by a constant function [duplicate]

Possible Duplicate: On distributions over $\mathbb R$ whose derivatives vanishes Why can I identify a distribution $G \in \mathcal{D}'((a,b))$, $\partial G = 0$ by a constant function?
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0answers
35 views

Show that continuous linear maps on the space of test functions take $C_K^\infty(\Omega)$ into some $C_{K_N}^\infty(\Omega)$

Let $\Omega$ be a nonempty open subset of $\mathbb{R}^n$, and let $\cup_{n=1}^\infty K_n = \Omega$ be an exhaustion of $\Omega$ by compact sets. Let $\mathcal{D}(\Omega) = \mathcal{D}$ be the standard ...
1
vote
4answers
36 views

Proof of the construction of Dirac Delta

The Dirac Delta function pops up in a wide variety of applications, especially in applications that require Laplace and Fourier transforms. But my question is: what's the proof that the distribution ...
0
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0answers
30 views

Limit of Distribution, Hilbert Transform

I want to peform a distributional limit of the following distribution: $\frac{2 i}{x^2 \epsilon} e^{-(t + x)^2/(4 \epsilon)} (F(\frac{t - x}{2 \sqrt{\epsilon}}) - F(\frac{t + x}{2 ...
0
votes
1answer
27 views

Is the product of a Schwartz function and a locally integrable function integrable?

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$
5
votes
1answer
98 views

$f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R) \implies |f(x)| \to 0$ as $|x| \to \infty$?

Suppose $f, \hat{f} \in L^{p}(\mathbb R) \cap C(\mathbb R)\cap L^{\infty}(\mathbb R), (1<p<\infty).$ My Question: Can we expect $\lim_{|x|\to \infty} |f(x)|=0$ ? (In other words, If $f$ and its ...
2
votes
0answers
117 views

Behavior of a Solution to Heat equation Compactly Supported in Time

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
2
votes
1answer
33 views

Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
2
votes
0answers
48 views

Is the following property of a Fourier Transform valid?

We know that $$\mathscr{F}\left\{f*g\right\}=\mathscr{F}\left\{f\right\}\mathscr{F}\{g\}$$ so I was wondering whether the inverse is true: ...
2
votes
0answers
79 views

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...