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).

learn more… | top users | synonyms

0
votes
0answers
20 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 ...
3
votes
3answers
46 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
51 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
140 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
16 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
30 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
vote
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, ...
2
votes
0answers
25 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
votes
0answers
45 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
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 ...
0
votes
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
votes
1answer
32 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
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
votes
1answer
51 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 ...
2
votes
1answer
50 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 ...
1
vote
0answers
13 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
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> = ...
1
vote
0answers
37 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
29 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
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 ...
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
votes
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 ...
1
vote
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 ...
0
votes
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 $$ \int |fg| <\infty. $$
2
votes
0answers
115 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
votes
0answers
47 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: ...
4
votes
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 ...
5
votes
1answer
97 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 ...
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 ...
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 ...
1
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
votes
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 ...
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 ...
1
vote
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 ...
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 ...
4
votes
0answers
77 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 = ...
3
votes
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 ...
0
votes
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 ...
0
votes
0answers
40 views

Tensor product of the Heaviside distribution

I would like to prove that: \begin{equation} H_{(a,b)}=H_a \otimes H_b \end{equation} So far I have: \begin{equation} \langle H_a(x) \otimes H_b(y), \phi\rangle=\langle H_a(x),\langle ...
4
votes
2answers
140 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 ...
0
votes
1answer
39 views

Prove if $T\in\mathcal{D}'(\mathbb{R})$ and $\mathrm{d}T/\mathrm{d}x=0$ then $T$ is the constant distribution.

Prove if $T\in\mathcal{D}(\mathbb{R})$ and $\mathrm{d}T/\mathrm{d}x=0$ then $T$ is the constant distribution. I am having some problems both proving this problem, as well as understanding ...
1
vote
0answers
21 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 ...
3
votes
2answers
41 views

Pointwise convergence and the convergence of a distribution sequence.

The sequence of functions $f_n(x)=\{tanh(nx)\}_{n=0}^{\infty}$ does not converge uniformly to $f(x)=sgn(x)$ but only pointwise. Is it then, still possible that the sequence of distributions ...
-1
votes
1answer
26 views

Proof that the distibution sequence $\int_{0}^{\infty}(tanh(nx)-1)\phi(x)dx \to 0$

Let $T:\phi \mapsto \langle T_f,\phi\rangle =\int _{-\infty}^{\infty}f(x)\phi(x)dx$, where $\phi(x)$ is a test function, be a distribution. I would like to prove that the sequence of distributions: ...
0
votes
0answers
23 views

A question involving weak convergence in $\sigma(L^1, L^\infty)$

Exercise 4.15 from H. Brezis - "Functional analysis..." Let I = (0, 1) and $f_n = n e^{-n x}$ a sequence of functions. Show that $$ f_n \;\; \text{does not converge weakly to} \;\; 0 \;\; \text{in} ...
0
votes
0answers
22 views

How to check a linear map between topological space is continuous?

I am reading something about distributions, and I have a question. I think it is not hard, but I don't know how to explain it rigorously. Suppose $M$ is a smooth manifold, $V$ is a Fréchet space, and ...
0
votes
1answer
33 views

Order of a distribution and its derivatives

For $\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$ , define $u(\varphi):=\int\partial^{\alpha}\varphi(x,0,0)dx$ for some multiindex $\alpha$ . It's pretty clear to me that $u$ is a distribution. ...