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Questions tagged [fourier-transform]

continuous Fourier transform, discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST)

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1answer
41 views

Proof of $\int_{-\infty}^{\infty} e^{-\pi t^{2}} e^{- i 2 \pi t v} dt = e^{- \pi v^2}$ using binominal square

I tried to proof this as a (signal engineering) homework using binomal square, but the example answer was given using differential equations. I'd like to know if my approach was possible. I tried ...
2
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0answers
24 views

Using Riemann-Lebesgue lemma

I am reading a paper by VAALER. He is using Riemann-Lebesgue lemma and saying that below function tends to $0$ as $N \to \infty$ $$\int_{-2}^{2} \frac{\pi t}{\sin \pi t}(\cos \pi(2N+1)t)e^{2 \pi i t ...
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0answers
32 views

Fourier transformation and eigenvalues

Let $a,b>0$, $F$ be Fourier transformation and $\chi$ be indicator function. Suppose $T:L^2(\mathbb{R}) \to L^2(\mathbb{R})$ $Tf(y):=F^{-1}(F(f\chi _{[-a,a]})\chi _{[-b,b]})$ and $\{c_n\}_{n=1}^{\...
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0answers
18 views

The Inverse Fourier Transform $\mathcal{F}^{-1}(|\xi|^k)\in L^1$

I was trying this question asked earlier today: Inverse of a Fourier Transform is $L^1 \cap L^{\infty}$ and I got the answer as long as I can show one thing: $\mathcal{F}^{-1}(|\xi|^k)\in L^1(\...
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0answers
20 views

using the Fourier transform, solve

using the Fourier transform, solve $$\dfrac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}},\hspace{3mm}-\infty<x<\infty,\hspace{2mm}t>0 \\u(x,0)=\left\{\begin{array}{lll}u_{0},&...
3
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2answers
62 views

Inverse of a Fourier Transform is $L^1 \cap L^{\infty}$

I was reading a Navier-stokes paper by Kato, and he affirm that if $k\geq0$ than $$\mathcal{F}^{-1}(| \xi |^ke^{-|\xi|^2})\in L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$$ (here $\mathcal{F}^{-1}$ ...
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1answer
45 views

Inverse Fourier Transform of the Fourier Transform

The FT of a function of time $f(t)$ transforms the function from the time domain to the frequency domain, such that: $$\mathscr{F}[f(t)]=F(\omega)=\int ^{\infty}_{-\infty}f(t)e^{-i\omega t}dt$$ And ...
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0answers
26 views

Solution for the Fourier transform of multiplication of two error functions.

I'm hoping someone can help me with what a Fourier transform problem. I seek the Fourier transform of $f(t)$, where: $$f(t) = a\left(1+\mathrm{erf}\left(\frac{\ln(t)-u_1}{\sigma_1\sqrt{2}}\right)\...
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0answers
41 views

Fourier transform of product of Bessel functions

I need help finding the Fourier transform of the function $$ \rho(\vec{r}) = \alpha \delta_{\vec{r},0} \left(\lambda\lambda' J_1 (\beta |\vec{r}|)Y_1(\beta |\vec{r}|) - \pi^2 J_0 (\beta |\vec{r}|)Y_0(...
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39 views

Demonstration to check basis for fourier series

I have an issue about a question posted on another forum. The user gatsu on that forum posted (originally in French) that starting from the following hermitian inner product on periodic ...
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0answers
39 views

Fourier transform of the ln of a variable

Consider the following Fourier transform: $$\hat{f}(\ln \xi) = \int_{-\infty}^{+\infty} f(x) e^{-2\pi i x \ln{\xi}} \, \, dx $$ Assuming I can calculate $\hat{f}(\ln \xi)$, how can I go from $\hat{f}(\...
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0answers
13 views

What is principal value in delta function integral? [on hold]

The delta function may have different forms of definition. One related to Fourier transform is shown below, $$\int_{-\infty}^{\infty}\!dt ~e^{i\omega t}~=~2\pi\delta(\omega).$$ then I wonder what if ...
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2answers
70 views

Identifying $\int_{-\infty}^\infty e^{i k x} dx$ as Dirac delta distribution

The expression $\int_{-\infty}^\infty e^{i k x} dx$ is sometimes identified as the Dirac delta function. This identification is said "formal" or "symbolic", and some physics texts say that the theory ...
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1answer
45 views

Calculate a Fourier transform

I need help to calculate the Fourier transform of this function: $f(t)=\sin (e^{-at})$ Where $a$ is a constant. I tried to use the definition of the Fourier transform: $f(\omega)=\int_{-\infty}^\...
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0answers
48 views

Fourier transform of the projection operator in 3D

I have a vector field $A_i({\bf r})$, a Fourier transform given by $$ \tilde A_i({\bf k}) = \int d^3 r~e^{i {\bf r.k}}A_i({\bf r}),$$ and projections given by $\mathcal P_{ij}(\hat r) = \delta_{ij}...
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0answers
37 views

Calculate the Fourier transform

I need help to calculate the Fourier transform of this function: $f(t)=\sin (e^{-at})$ Where $a$ is a constant. I tried to use the definition of the Fourier transform: $f(\omega)=\int_{-\infty}^\...
1
vote
1answer
62 views

Fourier and Laplace transforms together, is this possible?

Answering on some posts on MSE about Laplace transform and Fourier transform I stumbled upon a question to which I cannot answer myself (not having a good ground in pure mathematics). The question ...
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1answer
45 views

Fourier transform of $\frac{2\sin(x)}{1+x^2}$

I'm trying to calculate FT of $$f(x)=\frac{2\sin(x)}{1+x^2}$$ First of all we have $$ \int_{-\infty}^{+\infty}\left|\frac{2\sin(x)}{1+x^2}\right| \, dx\leq 2\int_{-\infty}^{+\infty}\frac{1}{1+x^2} \, ...
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1answer
42 views

Find of Fourier transform [closed]

Find the Fourier transform of $f(x) = \{ (1- x^2) \text{ if } |x|<1 \text{and } 0 \text{ if } |x|>1 \}$ ? I tried but I cant find it.
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1answer
110 views

A list of proofs of Fourier inversion formula

The reason for this question is to make a list of the known proofs (or proof ideas) of Fourier inversion formula for functions $f\in L^1(\mathbb{R})$ (obviously adding appropriate hypothesis to get a ...
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1answer
51 views

Derive Fourier transform by analogy to Fourier series?

The Fourier series coefficients are often derived by assuming a function can be represented as a series $$f(x) = \sum_{n=0}^\infty A_n \cos\left(\frac{2\pi n x}{L}\right) + \sum_{n=0}^\infty B_n \sin\...
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3answers
43 views

Proving Continuous Fourier Transform Formulas

Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(...
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1answer
16 views

Analyticity of a Fourier transform

I was reading a physics paper and the author made a statement that I didn't follow. Suppose $p(t)$ is given by the Fourier transform of $\omega(E)$ \begin{equation} p(t) = \int_{-\infty}^{+\infty} ...
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2answers
176 views

Alleged trick for Fourier transform of $\frac{x^4}{1+x^4}$

I was supposed to evaluate the Fourier transform of the function $$f(x) = \frac{x^4}{1+x^4}$$ and apart from the standard way of integrating with complex contours, which I find pretty tedious, I ...
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1answer
38 views

Technique for Fourier transform

I was looking at this exercise Evaluate the Fourier transform of the following function $$f(x)=\frac{xe^{5ix}}{x^2+2x+10}$$ and the professor was using some techniques which I don't quite ...
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0answers
74 views

How can I solve the second order nonlinear PDE's?

I have three PDE's and I tried to solve it through Finite Fourier Transform (FFT) method but could not get the required result due to non-linear term in Eq. (1). How can I solve it? Is there any other ...
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0answers
31 views

Abstract of Fourier Transform

I am a bit confused on Fourier transform. I have an example that illustrates my understanding. Please conform me weather my understanding is correct or not. If i have a source that emmits the ...
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0answers
46 views

Inverse Laplace and Fourier Transform in Statistics

I am currently exploring the use of inverse laplace transform and inverse fourier transform in statistics. From what I have read, for a random variable $ X $ with $ f(x) $ and $ F(x) $ as its PDF and ...
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0answers
40 views

Prime gaps and gaps between successive critical zeros of zeta

Assuming RH, the sequence of critical zeros of the Riemann zeta function can be viewed as the Fourier transform of the sequence of primes. From a physicist point of view, the average gap between the ...
2
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0answers
77 views

Filling gaps in “a proof” of Fourier inversion formula

Suppose $f\in L^1(\mathbb{R})$. Define the Fourier transform of $f$ as: $$\hat{f}:\mathbb{R}\rightarrow\mathbb{C}, \xi\mapsto\int_{\mathbb{R}}f(t)e^{-2\pi i\xi t}\operatorname{d}t.$$ Suppose that $\...
0
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1answer
71 views

Every function can be represented as a Fourier Series - but why?

I can't really find a good answer to this question - the statement just seems to be assumed everywhere I look. Admittedly, I am not too well versed in the topic but I am an engineer and can understand ...
1
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1answer
28 views

Fourier transform of Gaussian is equal to derivative of Fourier transform of Gaussian times constant

I'm working on a few problems in Reed & Simon, and I ran across this problem. Compute the Fourier transform of $f(x) = e^{-\alpha x^2/2}$ via the following steps. (a) Prove that $-\lambda \hat{f}(...
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1answer
27 views

Solution for Heat equation like Boundary problem solved with Sine or Cosine transform

I have this problem: $$ \left\{\begin{matrix} u_{xx}= u_{t}+u && 0<x< +\infty & t>0\\ u(x,0)=0 &&& x>0&\\ u_{x}(0,t)=f(t)&&& t\geq 0 \end{...
2
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1answer
42 views

Problem using Parseval's theorem for solving an integral

I need to use Parseval's theorem to calculate the following integral: $$\int_{-\infty}^{\infty}\left |\frac{1-e^{-iwt}}{iw} \right |^{2}dt$$ I thought to find the transform of $$f(t) = \frac{1-e^{-...
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3answers
204 views

Fourier transform derivation

I'm reading Hassani's Mathematical Methods book specifically the chapter on Integral Transforms. He derives the fourier transform starting with the concept that the fourier transform has a kernel of ...
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0answers
30 views

Parseval equality with $\int |f|^{1}$

The famous Parseval equality states that \[ \int|f|^{2} = \int |F|^2, \] where $f$ and $F$ are related to one another by \[ \int f e^{-2 \pi i r y } dr = F(y). \] My question is, whether there is any ...
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0answers
22 views

Fourier Transform of Manchester (twinned-binary) function

Given the following function used in binary telecommunications: I'd like to know how I would obtain $P(f)$ the Fourier transform of this function. The function can be written as: $$ p(t) = \left\{ ...
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1answer
25 views

Given $a>0$, $\frac{1}{x^2+a^2}$ is not a Schwarz function.

Given $a>0$, $f(x) = \frac{1}{x^2+a^2}$ is not a Schwarz function. Please verify if this is correct: Although Poisson Sumation formula is working for this function $f$, I think it is not Schwarz,...
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1answer
29 views

Derivation of a fourier transfrom of a derivative multiplied by a linear function

How can the following identity be derived $$F\left\{\frac{d(xf(x))}{dx}\right\}=-k\frac{d\hat{f}(k)}{dk}$$ where $$\hat{f}(k)=F\left\{f(x)\right\}=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx$$...
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1answer
33 views

How to interpret the results of a Discrete Fourier Transform

I'm trying to understand the Discrete Fourier Transform so that I can understand the Fast Fourier Transform so that I can write a program to calculate it. I know that there are existing libraries that ...
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1answer
54 views

What's the name of the Fourier “identity” $\frac1{2\pi}\int_{-\infty}^\infty\int_{-\infty}^\infty\,dk\,dx\,e^{i(k+k')x}\tilde\psi(k')=\tilde\psi(-k)$?

I came across a property of the Fourier transform (as shown in the picture), and I am having trouble finding what theorem or identity it is. I have tried searching various source but had no luck. ...
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0answers
47 views

$2$-Norm for Convolution [duplicate]

Let $C_c(\mathbb{R})$ be the following: $$C_c = \{ f \in C(\mathbb{R}) \mid \exists T > 0 \text{ s.t. } f(t) = 0 \text{ for } |t| \geq T\}$$ Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator ...
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0answers
40 views

Doubts about Fubini

I am reading about Fourier Transform on $S(\mathbb{R})$ to have a first formal approach. The book I read is 'Fourier Analysis' by Rami Shakarchi and Elias.M.Stein. In the chapter about Fourier ...
2
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2answers
33 views

Prove $\lim_{|\alpha |\to \infty }\hat f(\alpha )=0$

Let $f\in L^1(\mathbb R)$. Set $$\hat f(\alpha )=\int_{\mathbb R}f(x)e^{-2i\pi x\alpha }dx.$$ Prove $$\lim_{|\alpha |\to \infty }\int_{\mathbb R}f(x)e^{-2i\pi x\alpha }dx=0.$$ The thing is even if I ...
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3answers
52 views

Compute the Fourier transform of a $L^2$ function.

Let $f\in L^2(\mathbb R)$. Is it possible to compute the Fourier transform of an $L^2(\mathbb R)$ function ? I'm asking this question because in a book I'm reading, they take the Fourier transform of ...
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2answers
46 views

Regarding Fourier transform

Let $f\in L^1(\mathbb{T})$. Where $\mathbb{T}$ is the unite cirle in the complex plane. Define $$g(e^{it})=f(e^{2it}).$$ Show that $\hat{g}(n)= \hat{f}(n/2)$ if 2 divides $n$ and $\hat{g}(n)= 0$ if 2 ...
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1answer
72 views

Inverse Fourier transform of Lorentzians and sign function

I'm trying to caculate the inverse Fourier transform of $$ G(\omega) = \dfrac{(\omega+a)^2+b^2}{((\omega-c)^2+b^2)((\omega+c)^2+b^2)} \mathrm{sgn}(\omega-d)$$ It is the product of two Lorentzians and ...
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2answers
54 views

Can I change the limits in the Fourier transform definition of the Dirac delta function?

The Dirac delta function is often defined as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i p x} dp$$ Is there a way in which $$\delta(x)=\int_0^\infty e^{ipx} dp$$ is also correct? For ...
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0answers
23 views

fft of sawtooth signal

I have a very basic question. sorry for such a naive question! I have a sawtooth signal which look like this: this is how I generated this signal : ...
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0answers
15 views

Boundedness of Fourier transforms for Battle-Lemarie wavelets

Let $\phi$ and $\psi$ be the scaling function and wavelet for the Battle-Lemarie wavelet of order $n$. Is it true that $$ \sup_{x \in \mathbb{R}} |\hat\phi(x)| < \infty, $$ $$ \sup_{x \in \mathbb{...