1
vote
1answer
22 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
1
vote
0answers
18 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
0
votes
2answers
46 views

Calculating $\int_{-\pi}^{+\pi} e^{ixt} e^{-i \omega t} dt$

We know that Fourier Transform of $e^{ixt}$, where $x$ is a real parameter, $t\in \mathbb R$ is $$\int_{-\infty}^{+\infty} e^{ixt} e^{-i \omega t} dt=\int_{-\infty}^{+\infty} e^{ixt-i \omega t} ...
2
votes
0answers
65 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
2
votes
1answer
26 views

fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
0
votes
0answers
14 views

radon transformation backprojection

I am working on image recontruction and I try to find out how the radon transformation works. I have benn using mainly Natterer, F. and Wubbeling, F.: Mathematical Methods in Image ...
2
votes
0answers
53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
5
votes
2answers
52 views

Computation of the fourier transformation of a function with a matrix

I want to compute the Fourier transformation of the following function: \begin{align} f:& \mathbb R^n \rightarrow \mathbb R \\ & x \mapsto \exp(-\left<Ax,x\right>) \end{align} where ...
1
vote
1answer
27 views

Does the integral in the formal 2D Fourier transform of the logarithm converge?

If $k$ is a nonzero vector in $\mathbb R^2$, how to interpret this integral: $$\int_{\mathbb R^2}e^{ik\cdot x}\ln{|x|}dx$$ Does it converge and in what sense? Thanks in advance.
-2
votes
2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
0
votes
2answers
26 views

$\int\exp(-jnw_0t)\,dt$ integral calculus.

I seem to forgot these parts of integral calculus. I am trying to determine the Fourier coefficient in complex exponential form. Here, $t$ is the variable being integrated and $n$ is for all ...
3
votes
2answers
91 views

What does the Fourier transform of $1/x^2$ mean?

If I ask Mathematica to compute the Fourier transform of $\frac{1}{x^2}$ using the FourierTransform function, it gives me a result of ...
3
votes
1answer
33 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
3
votes
2answers
97 views

Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
2
votes
1answer
24 views

Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
1
vote
0answers
62 views

Easiest way to prove integral of $e^{ikx}$ is $\delta(k)$

What is the easiest way to to derive the following equation: $$\int_{-\infty}^{\infty}e^{ikx}dx = 2\pi\delta(k)$$ I understand the equation can be derived by assuming the Fourier integral theorem, ...
2
votes
0answers
22 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
0
votes
1answer
91 views

Integral is equal to $0$

Let be $f \in L^1[0,1]$, then it applies $ \int_0^1 \exp(2i\pi xk)f(x n)\,dx=0$ for $n,k\in \mathbb{N}$ with $0<k<n$. Ideas: f can be extended to a function on $\mathbb{R}$ with period $1$, ...
1
vote
0answers
29 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
1
vote
0answers
145 views

Limit of an integration formula

Let $f$ be a smooth real (or complex) valued function defined on $S^2$. Then a direct calculation shows that $$\int_{S^2}f(x)e^{ixy}\, ...
1
vote
2answers
72 views

How to integrate this fourier transform?

I want to integrate $$\int_{-\infty}^{\infty} \frac{e^{itx}}{{1+x^2}} dx.$$ I don't see how substitution or integration by parts could help here. Does anybody know how to do this?
6
votes
3answers
50 views

Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$

I have the following problem: Be the equation: $$u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw$$ Show that $u\rightarrow 0$ as $x\rightarrow \infty$, even when $e^{-iwx}$ does not falter ...
1
vote
1answer
118 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
3
votes
1answer
205 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
3
votes
0answers
34 views

How do I tackle this integral: $\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$? Is my solution correct?

I want to solve the following integral: $$\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$$ I did the following: Substitute $\gamma(k) = k-k_0 \Leftrightarrow k = \gamma + k_0;~\gamma(\pm\infty) = ...
0
votes
1answer
30 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
3
votes
2answers
160 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
0
votes
0answers
16 views

Obtain the complex Fourier Series of the following function:

$$f(t)=t^3 \;\;\;\;\;\;\;\;\;\;\;\; -3/2<t\leq 3/2 $$ $$f(t)=f(t+3)$$ I've tried setting up an integral for $C_n$ coefficients using the formula $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ...
1
vote
1answer
47 views

Integral equation, Fourier transform

Find all functions $ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve $\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$, $ t\in \mathbb{R}$ How do I solve this? I know that the left part is the ...
0
votes
1answer
37 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
0
votes
2answers
50 views

How to calculate this basic Fourier Transform?

I am trying to calculate the Fourier Transform of $g(t)=e^{-\alpha|t|}$, where $\alpha > 0$. Because there's an absolute value around $t$, that makes $g(t)$ an even function, correct? If that's ...
0
votes
2answers
92 views

fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
1
vote
1answer
42 views

Integral from inverse Fouriertransform of 1/(1+p^2)^2

In a calculation I end up with the following integral $$\int_0 ^\infty \frac{p \sin (pr)}{(1+p^2)^2}dp , $$ could someone give me a hint how to evaluate this one? (This integral comes from the ...
0
votes
1answer
22 views

Existence of a subsequence and convergence to $0$ of function solving heat equation

Let $f^j(x)$ be a sequence of integrable functions on the circle such that $$\int_{-\pi}^{\pi}|f^j (x) |^2 dx = 1.$$ ALso, let $u^j(x,t)$ solve the heat equation on the circle with initial data ...
3
votes
0answers
58 views

Prove the converse of convolution theorem

I am trying to prove the converse of convolution theorem: $$ \mathscr{F}[f(x)g(x)]=\frac{1}{\sqrt{2\pi}}\,\widetilde{f}(\omega)*\widetilde{g}(\omega)$$ I try to apply the definition of convolution ...
0
votes
0answers
18 views

How to find the inverse fourier transform of the fourier transform of $\delta(x-x_0)$ [duplicate]

I know that $F(\delta (x-x_0 )) = e^{j\omega x_0} $ and so therefore $F^{-1}(e^{j\omega x_0 }) = \delta(x-x_0) $ but when I try to do the integral for the inverse fourier transform: I get ...
1
vote
2answers
73 views

Fourier transform of function

What is Fourier transform of $$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$ I tried to calculate it using $$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$ and $$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$ and ...
18
votes
1answer
328 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
0
votes
0answers
179 views

What's the Fourier Transform of an Error Function?

What is the Fourier transform of $\displaystyle \operatorname{Erf}\left[a+bx^{2}\right] $? I need this in order to evaluate $$ \int_{-\infty}^{\infty}e^{-\beta x^{2}}Erf\left[a+bx^{2}\right]dx $$ ...
2
votes
0answers
37 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
1
vote
0answers
63 views

On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$ Can we use residue formula?
1
vote
1answer
101 views

Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
2
votes
2answers
61 views

How can this integral be rewritten with convolutions?

I've got $f:\mathbb{R}\rightarrow\mathbb{R}$ bounded and I'm trying to write `$\mathtt{f}$,' a discrete version of $f$, where each element in the domain takes on the average of the corresponding ...
4
votes
2answers
145 views

Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
4
votes
1answer
139 views

A Parseval-like theorem for Mellin transforms

A particular case of Parseval's theorem for Fourier transforms says that if $f$ is square integrable on $\mathbb{R}$, then $$ \int_{-\infty}^{\infty} |f(t)|^{2} \ dt = \int_{-\infty}^{\infty} ...
1
vote
1answer
32 views

time integration property of fourier transform

I am having some trouble with some Fourier transform, Suppose that $F(\omega)$ is the Fourier transform of $f(x)$, i.e. where $$F(\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}\,dx.$$ What is ...
2
votes
1answer
44 views

Riemann-Lebesgue application

By the Riemann-Lebesgue lemma, I have shown that for any finite interval measurable set $I$ of finite measure, any $h \in \mathbb{R}$, $$\lim_{n \to\infty}\int_I \cos (n(x+h)) \mathop{dx} = 0.$$ I ...
0
votes
0answers
16 views

Relating Fourier Transform to an Integral involving Sin(vt)

I have data for a function $S(Q)$ and I'm trying to find values for a different function $g(r)$ Now I know $g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ$ This is closely related to the sine ...
0
votes
1answer
58 views

Inverse Fourier transform problem

Can anybody please guide me how to compute the following inverse Fourier Transform ? $$ p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{1}{(1-j\omega\bar{x})^K}e^{-j\omega x}d\omega $$ I shall ...
0
votes
1answer
48 views

Fourier transform real and imaginary part question?

I have to find the fourier transform of $f(t)=e^{-a^*t}*u(t)$ For a>0 the signal has an infinite value therefore doesnt have a Fourier transform.For a>0 we have: ...