# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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} ...
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 ...
### 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: \$...