Linked Questions
18 questions linked to/from Fourier Transform of Derivative
3
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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 ...
- 167
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
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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 ...
- 693
1
vote
2
answers
695
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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) =...
- 649
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
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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 ...
- 25
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
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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
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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)}.
$$
How do I now use this to find the ...
- 113
2
votes
0
answers
185
views
Fourier transform on Green's function
I need to do a Fourier transform for the next Green's function:$F[(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}-m^{2})\cdot G(x,x^{'})]$.
My solution is: $\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{...
- 157
2
votes
2
answers
110
views
Derivative of a function and Sobolev space
Let function $f$ belongs to the Sobolev space or order $\beta$ defined by
$$
\mathcal{S}^{\beta}(\mathbb{R}) = \left\{u \in L^2(\mathbb{R}): \int_{\mathbb{R}}(1+|\xi|^2)^{\beta}|\hat u(\xi)|^2d\xi <...
- 21
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 ...
- 159
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-\...
- 13