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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1answer
50 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
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1answer
136 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
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0answers
15 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
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1answer
29 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 ...
1
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1answer
58 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
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1answer
107 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
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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 ...
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2answers
993 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
23 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 ...
2
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0answers
44 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
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0answers
31 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 ...
2
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1answer
481 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 ...
0
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1answer
20 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
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1answer
30 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
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0answers
32 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
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1answer
50 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 ...
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4answers
270 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
345 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
49 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
24 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
24 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 ...
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0answers
12 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 ...
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1answer
22 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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0answers
36 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
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1answer
28 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
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1answer
19 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
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1answer
383 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
34 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
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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 ...
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0answers
28 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
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1answer
25 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 $$ \int |fg| <\infty. $$
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1answer
96 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
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0answers
114 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
31 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
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0answers
46 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
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0answers
78 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 ...
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0answers
37 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
3
votes
2answers
48 views

Writing integral in terms of distributions

EDIT (now asking how to write $F$ as distributions, instead of writing the integral in terms of distributions): Let $F$ be the distribution defined by its action on a test function $\phi$ as ...
9
votes
1answer
184 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
2
votes
1answer
47 views

Prove that $\lim_{n\to \infty}\langle \operatorname{erfc}(-nx), \phi\rangle =\langle H_0, \phi\rangle $

Define the error function $\operatorname{erf}(x)$ as: \begin{equation} \operatorname{erf}(x):=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-y^2}dy \end{equation} and ...
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vote
1answer
34 views

What's the difference between $C_c^{\infty}(\mathbb{R})$ and Schwartz Space for test spaces of distributions? Are there other spaces?

A distribution is an element of either $\left(C_c^{\infty}(\mathbb{R})\right)^{\ast}$ or of $S^{\ast}$, depending on literature. Why would we use one vs the other? What other spaces are there and ...
2
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1answer
41 views

Uniform convergence for the sequence $\psi_n(x)=n^{-1}e^{-n^2/(n^2-x^2)}$

How can I prove that the sequence of functions: \begin{equation} \psi_n(x)=\begin{cases} n^{-1}e^{-n^2/(n^2-x^2)}, & |x|\leq n \\ 0, & |x|\geq n \end{cases} \end{equation} convergences ...
3
votes
1answer
158 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
0
votes
1answer
50 views

Problem with second distributional derivative

I have the following function: $ f(x) = \begin{cases} \sqrt{x}, & \text{if $x>0$} \\ \sqrt[3]{|x|}, & \text{if $x<0$ } \end{cases} $. I have to find $f'(x)$, $f''(x)$ as ...
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0answers
24 views

Example concearning the Schwartz Kernel Theorem?

Let $\Omega_i\subseteq \mathbb R^{n_i}$ ($i=1, 2$) be open subsets. Given $K\in\mathscr{D}^\prime(\Omega_1\times \Omega_2)$ one can obtain a continuous linear operator ...
4
votes
2answers
136 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where ...
3
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1answer
120 views

Convergence to the Dirac Delta Function

Let $h\colon[0,1]\to \mathbb{R}^+$ be any bounded measurable non-negative function with a unique maximum at $a$ and $h$ is continuous at $a$. For $\lambda>0$ define $h_\lambda(x)=C_\lambda ...
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0answers
19 views

Deriving existence of classical Fourier transform via the space of temperate distributions

If for some measurable function $f:{\bf R}^n\rightarrow{\bf R}$ the functional $h\mapsto\int fh$ is in ${\scr S}'$ (space of temperate distributions) and there exists some measurable $g$ such that the ...
4
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0answers
76 views

How to prove that this solution of heat equation is not a tempered distribution?

A theorem of PDEs sais that the following Cauchy problem for the heat equation \begin{align*} & \partial_t u = \partial_{x}^2 u, \quad (t,x) \in \mathbb{R_+} \times \mathbb{R}, \\ & u|_{ t = ...
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0answers
31 views

Differential Equation of a Distribution

How to solve the differential equation in $\mathcal{D'}(\mathbb{R})$: \begin{equation} T''=\delta_a \Leftrightarrow \langle T'',\phi\rangle=\langle \delta_a,\phi''\rangle, \forall \phi \in ...