-1
votes
2answers
54 views

Check: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. [on hold]

Part A: Find the Fourier transform of $f(x)=ax+b$ with $a,b \in \Bbb R$. Part B: Generalize the previous problem and deduce a formula for the Fourier transform of a polynomial of degree m. ...
0
votes
2answers
41 views

Let $\langle S, \psi \rangle=\sum_{n \in \mathbb N} \int_0^n \psi'(x)dx$. Is S a distribution?

Let $\langle S, \psi\rangle=\sum_{n \in N} \int_0^n \psi'(x)dx$. Is S a distribution? I claim that S is not a distribution. I know that if S was a distribution it would satisfy the following ...
1
vote
1answer
49 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
0
votes
1answer
20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
2
votes
1answer
36 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...
1
vote
0answers
45 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
1
vote
1answer
56 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
0
votes
1answer
57 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
0
votes
0answers
32 views

Eigenvectors of Laplace operator on a particular functiona space

Exercise 8.15 in [1] is: "Show that $\Delta u=\lambda u$ has no solutions of polynomial growth if $\lambda > 0$, but does have such solutions if $\lambda < 0$." How should I make sense of this? ...
1
vote
1answer
57 views

Fourier transform of $\frac{x_i^2}{|x|^2}$

For a function $f$ in $L^1(\mathbb{R}^n)$, it is natural to define the Fourier transform as $$\mathscr{F}(f)(\xi)=\int_{\mathbb{R}^n}f(x)e^{-ix\cdot \xi}dx.$$ And the we may extend it to rapidly ...
0
votes
0answers
24 views

How to interpret a Fourier transform interms of distribution? [duplicate]

In Folland's Real Analysis on p. 251, it is defined $$ \breve{f}(x)=\hat{f}(-x) $$ With this definition, if we go to p. 276 there is $[\frac{\text{sin} 2\pi t|\xi|}{2\pi|\xi|}]^{\breve{}}$. But I ...
0
votes
1answer
46 views

Computing Fourier transform of a surface measure.

I am almost beginner in the topics of Fourier transform. So, I am asking this question here. Let $n=3$ and let $\mu_t$ denote surface measure on the sphere $|x|=t$. Then how do we show that $$ ...
1
vote
1answer
34 views

Holomorphic Schwartz-space-valued function

I was trying the last few days to generalize the Paley-Wiener theorem in a quite obvious direction... or so I thought. The original Paley-Wiener theorem talks about functions and distributions with ...
1
vote
0answers
41 views

Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
0
votes
1answer
19 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
votes
1answer
33 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
2
votes
1answer
51 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
2
votes
1answer
34 views

What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?

If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
6
votes
1answer
75 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
2
votes
1answer
52 views

Relationships between growth rates of a distribution and smoothness of its Fourier transform

Let $f\in \mathcal{S}^\prime(\mathbb{R})$ be a tempered distribution, and $\hat{f}$ be its Fourier transform. It is known that when both $f$ and $\hat{f}$ are $L^2$ functions, there are relationships ...
2
votes
1answer
172 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 ...
1
vote
1answer
60 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...
0
votes
0answers
18 views

Boundary of real part of functions in $H^p$ and Poisson nontangential maximal function

I have two questions when reading on $H^p$ spaces, many books do not give their proofs. First we reminde that $H^p(\mathbb R^2_+)$ consists of all functions $F$ which is analytic in the upper half ...
0
votes
2answers
134 views

Fourier transform of a unity function and of unit step function

Fourier transform of the unity function is the Dirac delta distribution. I think this means: In particular, the Fourier transform of the unity function is the Dirac delta distribution, ...
1
vote
0answers
44 views

What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
3
votes
0answers
60 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
1
vote
2answers
96 views

About a Fourier transform of a non- integrable function.

I'm trying to obtain the Fourier transform of the following function: $$F(x)=\frac{x}{1+x^2}$$ I have tried using Residue Theorem, but i think it can't be applied because the difference between the ...
1
vote
1answer
77 views

About convolution and Fourier transform

I have some doubts with this question: I we have $f,g\in\cal{S}$ (where $\cal{S}$ is the Schartz space) with $f\ast g=0$, Can we deduce that $f=0$ or $g=0$? What I did is apply Fourier transform, ...
1
vote
1answer
97 views

About the Fourier transform of the sign function

I'm trying to calculate the Fourier transform of the function $f(x):=sign(x)$. I have read some texts where this is solved approximating the function $f$ by other functions, $f_a$, defined as follows ...
0
votes
0answers
28 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
2
votes
0answers
57 views

Fourier analysis on bounded domain?

For tempered distributions on $\mathbb{R}^n$ we can write $\widehat{\nabla f}(p)=p\hat{f}(p)$ and hence by Plancherel, we have equations like $(\nabla f,\nabla g)=(p\hat{f}(p),p\hat{g}(p))$ for ...
0
votes
1answer
51 views

Confused by a proof in Strichartz' book on Fourier Transforms

Hi I'm confused by a proof on page 53 in Strichartz book on Fourier Transforms. Specifically, in the first equation on page 53, why is it valid to interchange the action of the distribution with the ...
3
votes
1answer
189 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
0
votes
0answers
37 views

Convolution Kernel..

does anyone what is the definition of the convolution kernel of an operator, $$A: C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n),$$ where $\mathcal{D}^{'}(\mathbb T^n)$ is the space ...
1
vote
1answer
146 views

Asymptotic behavior of Fourier transform

Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$, $f(x) = |x|^{-1}$. It is locally integrable, and its distributional Fourier transform is $F(f)(k) = g(k) = 4\pi/|k|^2$. Intuitively, the ...
1
vote
0answers
33 views

Fourier Transform of Non Tempered Function

I have a function $g(x) = (1 + x^{1/a} )^a$ which is not bounded and in fact does not even define a tempered function. However, I need to take the Fourier transform of this function. What I can ...
8
votes
3answers
408 views

Rigorous derivation/explanation of delta function representation?

I am interested in a derivation of the following representation for the Dirac delta function: $$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$ It is clear to me how the property ...
2
votes
0answers
88 views

What is the set of all functions which can be used as a 'convergence factor' for a Fourier Transform?

At times, I am required to take the Fourier Transform of some function that does not decay quickly enough for the Fourier Transform to converge in the usual sense. For example, $$ ...
7
votes
1answer
133 views

Fourier transform using principal value

Can anyone help me compute the Fourier transform of $ 1/|x|^{n-\alpha} $ in $\mathbb{R}^n $ where $ 0 < \alpha < n $ ? Somehow it becomes the principal value of $ 1/|x|^\alpha $ which I can't ...
1
vote
1answer
40 views

Small arguments of $f$ vs large arguments of $\widehat{f}$

Say I know the behavior of $f:\mathbb{R} \to \mathbb{R}$ in the vicinity of 0. Are there any results linking that to the behavior of its Fourier transform $\widehat{f}(\xi )$ for large values of $|\xi ...
2
votes
1answer
127 views

Distributions as derivatives

Please help me to understand Step (2),(3),(4) and (6) of this theorem. I know mean Value theorem of calculus and here in first step they use it. This is something that mvt implies and I don"t know how ...
8
votes
2answers
175 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
2
votes
1answer
161 views

Distribution with finite support.

If $f \in\mathcal D′(\Bbb R^n)$, is a distribution and support of this distribution is a set of finite points in $\Bbb R^n$. Can anyone tell me that what will be the general form of this ...
3
votes
1answer
80 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
3
votes
2answers
203 views

Fourier transform

I am new to the distribution theory and have some difficulties to calculate curtain fourier transforms. Can you help me with $$\frac{e^{-xb}}{x+i0}$$ I got to the point $$\lim_{\epsilon \to ...
2
votes
0answers
97 views

Fourier transform of a tempered distribution

anybody knows how to calculate the fourier transform of $e^{-ax}, a>0$ in the sense of tempered function. I manage to find out it is $ \delta $ (y+ia) but it does not seem right as the 'argument' ...
1
vote
1answer
305 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
3
votes
2answers
100 views

Problem of convolution.

If we are given with a polynomial $\mathcal P$ and a compactly supported distribution $g$. Can we prove that their convolution will be a polynomial again?
3
votes
1answer
92 views

prove $ \mathcal F(f) = c_1\delta + c_2 \delta' + c_3\delta'' + T_g $ with $f(x)=|x^2 -1|$

let $f(x)=|x^2 -1|$ be a tempered distribution (i proved it) , and calculated its 3rd derivation (as a distribution) and then this stopped me : prove that we have : $$ \mathcal F(f) = ...
2
votes
1answer
70 views

prove the existence of $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $

prove that it exist $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $ with $\mathcal F$ fourier transform , $f\in \mathbb D(\mathbb R)$ , $\delta$ ...