# Tagged Questions

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### Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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### An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty$. 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 ...
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### 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 \lim_{t\to ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ... 2answers 109 views ### About a Fourier transform of a non- integrable function. I'm trying to obtain the Fourier transform of the following function:$$F(x)=\frac{x}{1+x^2}$$I have tried using Residue Theorem, but i think it can't be applied because the difference between the ... 1answer 82 views ### About convolution and Fourier transform I have some doubts with this question: I we have f,g\in\cal{S} (where \cal{S} is the Schartz space) with f\ast g=0, Can we deduce that f=0 or g=0? What I did is apply Fourier transform, ... 1answer 163 views ### About the Fourier transform of the sign function I'm trying to calculate the Fourier transform of the function f(x):=sign(x). I have read some texts where this is solved approximating the function f by other functions, f_a, defined as follows ... 0answers 32 views ### u \in H^n then also |u|^2 \in H^n for complex valued function in Sobolev space Suppose that a complex valued function u is in the Sobolev space H^n(R^n) = W^{n,2}. Is it necessarily true that |u|^2 \in H^n ? The result is true for real - valued functions since we know ... 0answers 63 views ### Fourier analysis on bounded domain? For tempered distributions on \mathbb{R}^n we can write \widehat{\nabla f}(p)=p\hat{f}(p) and hence by Plancherel, we have equations like (\nabla f,\nabla g)=(p\hat{f}(p),p\hat{g}(p)) for ... 1answer 52 views ### Confused by a proof in Strichartz' book on Fourier Transforms Hi I'm confused by a proof on page 53 in Strichartz book on Fourier Transforms. Specifically, in the first equation on page 53, why is it valid to interchange the action of the distribution with the ... 1answer 196 views ### Exercises about Distributions I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ... 0answers 39 views ### Convolution Kernel.. does anyone what is the definition of the convolution kernel of an operator,$$A: C^\infty(\mathbb T^n)\rightarrow \mathcal{D}^{'}(\mathbb T^n),$$where \mathcal{D}^{'}(\mathbb T^n) is the space ... 1answer 155 views ### Asymptotic behavior of Fourier transform Consider the function f:\mathbb{R}^3\rightarrow \mathbb{R}, f(x) = |x|^{-1}. It is locally integrable, and its distributional Fourier transform is F(f)(k) = g(k) = 4\pi/|k|^2. Intuitively, the ... 3answers 140 views ### The inverse Fourier transform of 1 is Dirac's Delta From the definition of the Dirac delta \delta_0 one can infer that its Fourier transform is identically equal to 1. But going in the other direction is not as straightforward. How can one show ... 0answers 34 views ### Fourier Transform of Non Tempered Function I have a function g(x) = (1 + x^{1/a} )^a which is not bounded and in fact does not even define a tempered function. However, I need to take the Fourier transform of this function. What I can ... 3answers 509 views ### Rigorous derivation/explanation of delta function representation? I am interested in a derivation of the following representation for the Dirac delta function:$$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$It is clear to me how the property ... 0answers 89 views ### What is the set of all functions which can be used as a 'convergence factor' for a Fourier Transform? At times, I am required to take the Fourier Transform of some function that does not decay quickly enough for the Fourier Transform to converge in the usual sense. For example,$$ ...
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 ...