Linked Questions

3 votes
1 answer
6k views

Fourier Transform of a Derivative [duplicate]

I'm trying to prove that: $$F\,\{f'(x)\} = -i\omega F(\omega) \qquad (1) $$ where $\, F(\omega) = F\,\{f(x)\}$ This is my procedure so far: $$F\,\{f'(x)\} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{...
  • 468
2 votes
1 answer
116 views

Fourier transform of $\partial_{x} u$ [duplicate]

I want to compute the Fourier transform of $\partial_{x} u(x)$ My definition is $\hat{u}(k)= \int_{\mathbb{R}}u(x)e^{-ik\cdot x}dx$. I am ignoring the constant from the definition. I am told the ...
28 votes
5 answers
6k views

Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$

I'm trying to prove the following claim: If $f$ is continuous and $\displaystyle\int_a^\infty |f(x)|\;dx$ is finite then $\lim\limits_{ x \to \infty } f(x)=0$. Here the counter example of all ...
  • 6,880
1 vote
1 answer
4k views

Fourier Sine Transform of the derivative of a function [closed]

I'm going over some old stuff of Fourier transforms, and came across the identity $\mathscr{F}_s[f']=-\omega\mathscr{F}_c[f].$ I know this is done using integration by parts but I'm having a problem ...
1 vote
2 answers
695 views

The Fourier transform of the derivative of a function $f\in L_1(\mathbb{R})$

Let $f\in L_1(\mathbb{R})$ (That is to say $f$ is absolutely integrable over $\mathbb{R}$) with derivative $f'\in L_1(\mathbb{R})$. The Fourier transform of $f$ is given by: \begin{align} \hat{f}(t) =...
4 votes
2 answers
700 views

Question about proof of Fourier transform of derivative

If $f\in L^1(\mathbb{R})$, $f'(x)$ exists and is continuous, and $f'\in L^1(\mathbb{R})$, then $\widehat{f'}(t)=2\pi i \widehat{f}(t)$. I've stated the above theorem from a textbook that I'm reading. ...
  • 641
1 vote
1 answer
474 views

Fourier transform of derivative when integrating by parts

As seen in https://math.stackexchange.com/a/430885/634773, we can find the Fourier Transform of the derivative of a function through the anti-transform. But if we do integration by parts, shouldn't ...
3 votes
0 answers
325 views

Fourier Transform of derivative a vector function

I am working on the proof of the Fourier Transform of the derivative of a function. I am accompanied by some proof lines but having some issue in one of the integral evaluation. I searched out its ...
  • 109
1 vote
1 answer
189 views

Heisenberg uncertainty principle from the integral of norm of schwartz function?

If we only have a schwartz function $f$: $\mathbb{R} \to \mathbb{C}$, that $\int_{-\infty}^{\infty} |f(x)|^2 \,dx = 1$, how can we show that $\int_{-\infty}^{\infty} x^2 |f(x)|^2 \,dx \cdot \int_{-\...
  • 751
1 vote
1 answer
177 views

Application of the operator $\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right)$

I need to apply the operator $$\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right) \tag{1} \label{1}$$ To the function $$M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) \tag{2} \label{2}$$ where $M(x)$ ...
1 vote
1 answer
129 views

How do I find the Fourier transform of $\mathcal{F}[\log(a^2+s^2)](s)$

For $a>0$ i have managed to show that this is the Fourier transform of the function. $$ \mathcal{F}[e^{-a|x|}](s) = \frac {2a}{\sqrt{2{\pi}}(a^2+s^2)}. $$ How do I now use this to find the ...
  • 113
2 votes
0 answers
185 views

Fourier transform on Green's function

I need to do a Fourier transform for the next Green's function:$F[(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}-m^{2})\cdot G(x,x^{'})]$. My solution is: $\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{...
2 votes
2 answers
110 views

Derivative of a function and Sobolev space

Let function $f$ belongs to the Sobolev space or order $\beta$ defined by $$ \mathcal{S}^{\beta}(\mathbb{R}) = \left\{u \in L^2(\mathbb{R}): \int_{\mathbb{R}}(1+|\xi|^2)^{\beta}|\hat u(\xi)|^2d\xi <...
0 votes
1 answer
130 views

Fourier transform of a function involving its derivative

I am trying to understand how to go about the following: We are given that the Fourier transform of a function $f(r)$ is $F(k)$ where $k$ is a representative wavenumber corresponding to some spatial ...
1 vote
1 answer
71 views

Fourier transform of the following time dependent expression

I am working with the following expression which describes a simple Markov birth-death model of particle transport of uniform diameter: \begin{gather*} \frac{\partial\eta(t)}{\partial t}=\nu+\mu\eta-\...

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