Questions tagged [fourier-transform]

The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

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15 views

Applying envelope through Inverse Fourier Transforms

I'm trying to make a function $\text{build}$ such that: $\forall s\in S, f\in F, \text{envelope} \in E :$ $$|\text{extract_frequency_component}(\text{FT}^{-1}(\text{build}(\text{envelope})(s)), f)| = ...
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Does there exists a non-zero smooth function on Torus, vanishing on an open set such that all its Fourier coefficients are non-zero?

Let us identify the torus $\mathbb{T}$ with $[-1,1).$ Does there exists a non-zero $f\in C^\infty(\mathbb{T})$, vanishing on an open set such that all its Fourier coefficients $a_n$ defined by $$a_n=\...
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49 views

Asymptotic behaviour of fourier transform of exponential of a polynomial

Consider the family of functions $f_n=e^{-x^{2n}}$, where $x$ is a real number. I am interested in the fourier transform $\hat{f_n}(t)=\int_{-\infty}^{\infty}f_n(x)e^{2\pi itx}dx$. While the exact ...
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A tricky integral using Fourier Transform and Dirac-functions

I need to calculate the following integral: $$ \boxed{I= \int_{0^+}^{t} \int_0^\infty f'(t')\, \omega^2 \cos(\omega(t'-t))\, d\omega\, dt'} $$ where $t>0$, $t' \in (0,t]$ and $f'(x)$ is the ...
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40 views

Proving $|x|^{\alpha} \in Wm,p(\Omega)$ [closed]

Verify that when $\Omega$ is bounded and $0 \in \Omega \subset \mathbb{R}^{d}$, $f(x) = |x|^{\alpha} \in W_{m,p}(\Omega)$ if and only if $(\alpha - m)p + d > 0$. $W_{m,p} (\Omega)$ ={$f \in L_p(\...
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50 views

Need help with Fourier transforms. Is the following true?

Let $f=p(x)e^{-x^2}$, where $p \in \mathbb{C}[x]$. I am trying to find out if the Fourier transform (FT) of $$\sum_{j=1}^n i^{j+k}\frac{d^k}{dx^k}(x^{j}\cdot f) \quad (\text{for some fixed } k \text{ ...
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Characteristic function and Fourier transform

Let $\phi_{x}(t)= E [ e^{itx}]$ be the characteristic function If X is a continuous random variable, then: $\phi_{x}(t)= E [ e^{itx}] = \int e^{itx} f(x)dx$ (being $f(x)$ the probability density ...
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What is the Fourier transform of functions of the type $p(x)e^{-x^2}$ $(p \in \mathbb{C}[x])$?

First some context to my question: I have two sets $M=\{p(x)e^{-x^2}:p\in \mathbb{C}[x]\}$ and $N=\{\hat{f}:f\in M\}$. Both are left modules of the Weyl algebra $A_1$. There are a few other technical ...
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71 views

Fourier transform of $\sqrt{f(t)}$

If the Fourier transform of $f(t)$ is $F(f)$, can you conclude that the Fourier transform of $\sqrt{f(t)}$ is $\sqrt{F(f)}$? Probably this is not always the case, but what are the cases in which this ...
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From continuous Fourier transform to discrete Fourier Transform

I am trying to derive the formula from continuous Fourier transform to discrete Fourier transform, however, I encountered a small problem. My derivations are the following. For a periodic function $f(...
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Fourier transform of linear combination of variable

Suppose I have a function $f(x,y)$ which satisfies a pde in $x,y$. Will following kind of Fourier transform will make any sense? $$\int f(x,y) e^{-i\omega(x-y)}d(x-y)$$ in other words if I take $u=x-y$...
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37 views

Inverse Fourier involving branch cut

I am looking for the closed form of the following integral involving inverse Fourier, $$\int_{-\infty}^\infty \frac{e^{-i\tau\omega}}{(A-\cosh(2\tau))^{\frac{\nu}{2}-1}}\,\mathrm{d}\tau$$ where $\nu&...
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The spectrum $m(f)$ of $m(t)$ , is an even triangle pulse , with bandwidth w , find m(t)

So I am trying to find the inverse fourier of m(f) by the definition. $$\begin{aligned}\int_{-\infty}^\infty m(f)e^{j2πft}df&=\int_{-w}^w m(f)e^{j2\pi ft}df\\&...
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$f(x) = 1 / \lvert x \rvert^2$, $x\in \mathbb{R}^3$ , for the Fourier transform F, prove by scaling: $ F(f) (y) = C \frac{1}{\lvert y\rvert}. $

Let $f(x) = 1 / \lvert x \rvert^2$, $x\in \mathbb{R}^3$, $\lvert x\rvert = \sqrt{x_1^2 + x_2^2 + x_3^2}$. Let $F(f)$ denote the Fourier transform of $f$. Assume that $F(f)$ is an $L^1_{loc}$ function ...
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How to find a positive function with compact support whose Fourier-Transform ist also positive

I'm looking for a non-trivial $L^1(\mathbb{R})$ function $f\ge 0$ with compact support and such that $$\hat f(\xi) = \frac{1}{(2\pi)^{\frac{d}{2}}}\int_{\mathbb{R}}f(x)e^{-ix\xi}dx\ge0$$ The last ...
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1answer
27 views

Fourier Transform of char. function of $d$-dimensional unit cube

I want to find the Fourier transform of the unit cube. So far, I have $$f(\xi) = \frac{1}{(2\pi)^\frac d 2}\int_{\mathbb{R}^d}\chi_{[-1,1]^d}e^{-i\langle x,\xi\rangle}dx = \frac{1}{(2\pi)^\frac d 2}\...
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How to interpret evaluating L(f(t))(σ+jω) for a specific σ for a *system*

The question is not about mathematics; that part is clear. I understand the relation between Laplace and differential equations and the multiplication/convolution. I have a system's impulse response f(...
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Multiplying tempered distributions by smooth functions

This is roughly exercise V.23 of Reed, Simon - Methods of Modern Mathematical Physics, vol. 1. Let $\psi$ be a $\mathcal{C}^\infty$ function such that for every multi-index $\alpha$ there exist $c_\...
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Summation of $\sum_{n=0}^{\infty}a^nq^{n^2}$

I am trying to find the result for the sum of the form $\sum_{n=0}^{\infty}a^nq^{n^2}$. The special case for $a=1$ is easily given by $\vartheta(0,q)$, where $\vartheta(z,q)$ is the third Jacobi Theta ...
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Find an explicit solution of the following system

I need to find an explicit solution $u(x,t)$ of the system $$\begin{cases} \partial_x^2 u = \frac{1}{k} \partial_t u \\ u(x,0) = f(x) \end{cases}$$ where $f(x) = x^2e^{-x^2}$, $\,x \in \mathbb{R}, \,t&...
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Does the operator $(\hat{f}\cdot m )^\vee$ maps Schwartz in it self?

Given $m \in L^\infty$ and $\phi \in \mathcal{S}$ a Schwartz function, is it true that $(\hat{f}\cdot m)^\vee$ is a Schwartz function?? I trying to prove this so I could conclude that operator of the ...
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Is solution multiplying Fourier transforms for product of two sequences wrong?

Consider the sequence $$ r[n] = \left\{ \begin{array}{ll} 1, & 0\leq n\leq M \\ 0, & \text{otherwise} \\ \end{array} \right. $$ for which $$ R(e^{j\omega}) = e^{-j\frac{M}{2}\...
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30 views

Fourier Transform of Product with Functional Phase

I would like to take the Fourier transform of a function of the form $$f(x) = e^{i\phi(x)}h(x)$$ Where $0 < C_1 \leq \phi(x) \leq C_2$ and $\phi \in C^\infty(\mathbb{R};\mathbb{R})$ and $h\in L^1(\...
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43 views

Examples of Measures that are Equivalent to their Self-Convolution

I'm interested in seeing (or generating) lots of examples of measures $\mu$ on $\mathbb{R}$ such that $\mu \sim \mu * \mu$. I'd love a reference (or even just a name) for these measures or, ...
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34 views

Proving that $\sin(x)\sin(nx)/x^2$ has $L^1$ norm tending to infinity

This is taken as a side question from Rudin's book on Real and Complex Analysis. I need to prove that $$f_n(x)=\frac{\sin{(x)}\sin{(nx)}}{x^2}$$ has an $L^1$ norm that tends to infinity as $n\to\infty$...
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25 views

L^1-Fourier Transform is not bounded below

I know that $\mathcal{F}_1:{\rm L}^1(\mathbb{R})\to {\rm C}_0(\mathbb{R})$ is not bounded below. I also know that since in ${\rm L}^2$ the operator is actually a diagonalizable unitary, I should not ...
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1answer
29 views

Fourier transform of trigonometric function

I would like to ask for some help on the Fourier transform of the following function. $F(t)=\frac{cos(\Omega t)}{(\lambda^2+t^2)}$ I can do the Fourier transformation with the cosine function. Thanks ...
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1answer
21 views

Fourier inverse transform of shifted cosine

Given the signal: $ cos(4 \pi f)e^{-j 2 \pi 5 f} $ I'm trying to apply the inverse fourier transform like this: $ \int^{+\infty}_{-\infty} cos(4 \pi f)e^{-j 2 \pi 5 f} \cdot e^{j 2 \pi f t} \, df $ $ \...
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31 views

$f \in L^1(G) \cap B(G) \implies \hat f \in L^1(\hat G) $

$G$ be a locally compact abelian group. $\hat G$ denotes the group of characters on $G$. $M(\hat G)$ be the space of regular complex Borel measure on $\hat G$. And $B(G):=\{f:G \to \mathbb C|\exists \...
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57 views

Compute integral involving modified bessel of second kind

I want to compute: $$I = \int_{0}^{\infty} x^2 [K_1(x)]^2 dx$$ where $K_1(x)$ is the Bessel Modified Function given by: $$ 2xK_1(x) = \int_{-\infty}^{\infty} \frac{e^{isx}}{(1+s^2)^{3/2}} ds $$ tip: ...
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How to show this function is in $L^1$?

$G$ be a locally compact abelian group. $\hat G$ denotes the group of characters on $G$. $M(G)$ be the space of regular complex Borel measure on $G$. Now let $\mu \in M(G)$ and $\hat{\mu}\in L^1(\hat ...
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1answer
30 views

Fourier coefficients of sin(1/k), i.e. essential discontinuity

Consider on $[-\pi,\pi]$ the function $$f(k) = sin \big(\frac{\pi^2} k\big).$$ This function has an essential discontinuity at $k=0$ and is smooth otherwise. My question is what are the Fourier ...
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14 views

Relationship between the Fourier transformation of functions and that of D-modules

In the theory of algebraic $D$-modules, the Fourier transformation is defined as an automorphism $\mathcal{F}$ of $n$-dimensional Weyl algebra $D_n := \mathbb{K}[x_1,\dots,x_n]\langle\partial_{x_1},\...
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25 views

Fourier transform of circulant or cyclic permutation matrix

I understand that a circulant is expanded as a polynomial in P $$C = C_{0} P + C_{1} P^{2} + \dots + C_{n} P^{n}$$ I also know that the columns of the Fourier matrix $F$ are the eigenvectors of $P$ ...
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1answer
25 views

EGC and waves $p-s$ for an earthquake: functions examples using Taylor's expansion

We know that the electrocardiogram (ECG) is a graphical representation of the electrical activity of the heart and in medicine plays an indispensable role. ECG is one of the indicators of the total, ...
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24 views

Inhomogeneous wave equation with periodic BC

I'm looking for the solution of the inhomogeneous 3D wave equation $\frac{\partial^2\rho}{\partial{t}^2} - c^2\nabla^2\rho = S(x,y,z,t)$ with periodic boundary condition in all three directions in a ...
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102 views

How to evaluate the Fourier sine transform of $1/x^3$

With the help of Maple, I have get the Fourier sine transform of $1/x^3,$ which is defined as $\sqrt{\frac{2}{\pi}}\int_0^{+\infty}\frac{\sin(x\omega)}{x^3}d x.$ And the output Maple given is ...
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1answer
157 views

Equality of Moment Generating Functions

Let $X,Y$ be be random variables whose moment generating functions $s\mapsto \mathbb{E}(e^{sX})$ exist and agree on either the interval $(-\delta,0]$ or on the interval $[0,\delta)$ for some $\delta &...
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8 views

low and high k damping for non-fourier methods

I'm working on solving a set of PDEs that describe micro-turbulence in a fusion plasma.There is an article with results that I'm trying to reproduce to make sure my numerical method is working ...
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1answer
52 views

Prove that Fourier Coefficients given by the orthonormal projection the closest element to v in W

Let $V$ be an inner product space of finite dimension. Let $v_1, v_2, ... , v_m$ be orthonormal vectors in $V$ and $W=\operatorname{sp}\{v_1, v_2, ... , v_m\}$. let $v$ be some vector and $\alpha _i=\...
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15 views

Fourier Transform on affine spaces

In order to understand Fourier transforms in more detail, I wonder how to define the Fourier transform on affine spaces. In other words: How to define the FT while maintaining a strict separation ...
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63 views

What is $\cos x-\cos2x+\cos3x-\cos4x…\pm\cos(Nx)$?

I want to arrive at a closed expression for $f_N(x)=\frac{2}{π}(\cos x-\cos2x+\cos3x-\cos4x...\pm \cos Nx)$ ($+\cos(Nx)$ if $N$ is odd, $-\cos(Nx)$ if $N$ is even) Using the fact that $\cos(x)=\frac{e^...
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21 views

Fourier transform of exp(-g(w)exp(-w))

I am looking for inverse fourier transform of $$e^{(-g(\omega)e^{(-a|\omega|)})}$$ I am mainly interested in $g(\omega)=b$, $g(\omega)=b\omega ^2$ and $g(\omega)=bi\omega$ Let me know if you need ...
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1answer
17 views

Compute integral of periodic function numerical spectrally

I have a $2\pi$-periodic function $f(x)$, and I want to calculate numerically the integral $\int_{0}^{\alpha}f(x)dx$ where $\alpha$ is a point in the interval $[0,2\pi]$. I have the function evaluated ...
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10 views

Fourier Transform for a variable dependent on another

I am dealing with finding the green function for a PDE where by change of variables I could make it linear. At first, I had a function of $f(x,y,t)$ which I change it to $f(x,z,t)$ considering $z=x^2+...
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2answers
67 views

Fourier transform of signum function

If we treat fourier transform as an operator on $L^1(\mathbb{R})$, then its image under fourier transform is the set of continuous functions which will vanish at infinity. It is well known that the ...
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36 views

Solving the equation $\frac{dx(t)}{dt} = -x(t)$ using a Fourier transform

I am trying to understand the frequency domain and Fourier transforms by using them to solve simple differential equations. In particular, I am interested in the equation: $$ dx(t) = x(t)dt \quad \...
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20 views

Fourier inversion formula for L^2 functions

I have got a question about the Fourier-inversion Formular. Given a function $f \in L^2(R)$ such that the following limes exists for almost every $x\in R$ \begin{equation} \lim_{N \rightarrow \infty} ...
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17 views

Uncertainty principle and support of functions

Let $B_1$ and $B_2$ be two closed ball with positive radius in $\mathbb{R}^d$. We know that if $\hat{f}$ is supported in $B_1$ then f cannot be supported in $B_2$. Do we have furthermore that there ...
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18 views

The DFT of $\sin(2 \pi j/N) + \cos(2 \pi j/N)$

So my question is: What is the DFT of $f_{j} = \sin(2 \pi j/N) + \cos(2 \pi j/N)$, for $j = 0, 1, \dots, N-1$? After writing $\sin$ and $\cos$ in the complex form, I arrived at the following answers: $...

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