Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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Fourier series with a weighted mean square norm

I am interested in Fourier series with a non-uniformly weighted error norm. What I mean by this is that the usual Fourier series of a periodic function is a minimizer of the mean squared error: $$ J_N ...
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Appropriate frequency domain transform for bounded periodic complex waves

I'm working on a wave inversion problem for which the data take the form of 2D spatial frequencies within boundaries. My understanding is that the appropriate frequency-domain transform for a periodic ...
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Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
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solving a complex fourier series

while solving for co efficient of a complex fourier series i came across this form cos(0.5*n*pi)+i*sin(0.5*n*pi) is there any way to simplify this? Note: N is an integer
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Characteristic Function as Fourier Multiplier

In lecture notes I have, it is mentioned that the characteristic function $\chi_I$ of an interval in $\mathbb{R}$ is an $L^p$ Fourier multiplier for $1 < p < \infty$. I thought this would be ...
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35 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 ...
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Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component. Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get ...
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(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
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fourier-transform: where is my mistake?

I am trying to do a fourier-transform of the function $$\psi(x,t=0) = \frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}e^{-\frac{x^2}{4\sigma^2}}e^{ik_0x}$$ My calculation is $$\int_{-\infty}^\infty ...
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conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
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Basic question about the Discerete Fourier Transfrom

I have trouble understanding the transition from the infinite integral of the Fourier transform $$ \mathcal{F}f(v) = \int^\infty_{-\infty}e^{ivk}f(k)dk $$ to the discrete version $$ \mathbf{F}f_n = ...
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$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
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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 ...
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Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
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Please help with this Discrete fourier transform question

Consider the ODE $\frac {d^2u}{dx^2} + 2\pi\frac {du}{dx} + \frac 54\pi^2u = g(x)$ where g is a periodic fuction with period 1 given by $g(x) = e^{\pi x}$ , $ 0 \le x \lt 1$. It is desired to find ...
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Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
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31 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 ...
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29 views

What can be observed by evaluating a polynomial at roots of order greater than the polynomial itself?

I have been reading through an algorithms book on the use of FFT for large number multiplication. An example it used to emphasize a point was: Evaluate the following polynomial at all roots of unity ...
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Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
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testing out the formula. I don't know how to intepret this!

$$\frac{d^{2}u}{dt^{2}}-4\frac{d^{3}u}{dt dx^{2}}+3\frac{d^{4}u}{dx^{4}}=0$$ $$u(x,0) = f(x)$$ $$\frac{du}{dt}(x,0) = g(x)$$ Relevant equations The attempt at a solution First I use fourier ...
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how to prove translation identity: $(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x)$?

Let $m, f\in S(\mathbb R^{n})$=The Schwartz space. My question: How to prove: $$(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x);$$ where ...
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properties of frequency- decomostion operator $\square_{k}^{\sigma}=\sum_{|\ell|_{\infty}\leq 1}\square_{k}^{\sigma}\square_{k+\ell}^{\phi}$?

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
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$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
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21 views

stroboscope effect

I have a disc with a line drawn on one of his radius that is turning with frequency $f$, and I want to sample the place of the line to find the frequency of the disc. So we know from the ...
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Nyquist Sampling Theorem

I'm working on the proof of the Nyquist sampling theorem. Mainly I'm wondering about the regularity conditions. In particular, suppose $f$ is continuous on $\mathbb{R}$ and $\hat{f}(k)=0$ unless ...
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Polynomial Factorization - 5 variables

Factorize a polynomial with 5 variables $P\in C[x_0,x_1,x_2,x_3,x_4]$, $$\sum^{4}_{j=0}x_j^5 + ...
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Fourier Analysis and applications to Abels groups

Find all functions $f:A\rightarrow C$ such that: $$\sum_{x\in A}|(f*f)(x)|^2 = |A|(\sum_{x\in A}|f(x)|^2)^2$$
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Derivative of function using discrete fourier transform (MATLAB)

I'm trying to find the derivative of a function $f(t)$ using the discrete fourier transform. My end goal is to do so numerically, but I suppose if I have the theory down then the rest should follow ...
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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Recommend Fourier Analysis Workbook or online examples

I am studying a graduate level course in Fourier Analysis, however my Functional Analysis background is extremely weak, I have also never met Lebesgue Integration and it has been a while since I ...
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Fundamental Solution to Laplace equation using inverse Fourier

I'm trying to do the following integral (which is an inverse Fourier transform): $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{e^{i (x_1\xi_1 + x_2\xi_2) }} {4\pi^2 (\xi_1^2 ...
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Parameter estimation using characteristic function

Is it possible to do parameter fitting using log-returns data & the characteristic function(CF) in Matlab? I have been trying it on the Variance Gamma Scaled Self-Decompasable (VGSSD) model CF for ...
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What do imaginary components of frequencies represent in Fourier?

I'm a Biology Ph.D. student with some math literacy (multivariable calc, discrete math, linear algebra, quantum chemistry) under my belt, so apologies if my knowledge is substandard for the work I'm ...
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38 views

Solve PDE with the Fourier transform

I have a problem with solving PDEs with the fourier trasform method when the function not depends only on x and t but also on the y variable. In particular, when I have to solve this equation ...
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Express the fourier coefficients of a autocorrelation function (complex form)

Here are some hints from the instructor: "Just plug the expression for the autocorrelation function into the formula for the Fourier coefficients, you get a double integral, and then smuggle in an ...
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Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
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Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases} $$ A problem in Stein's Fourier Analysis asks ...
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Reconstruction formula for a function of moderate decrease

I want to show that, for a function $f$ of moderate decrease and $\hat{f}(\xi)$ supported in $I=[-\frac{1}{2},\frac{1}{2}]$, $$f(x) = \sum^{\infty}_{n=-\infty} f(n)K(x-n)$$ where $K(y) ...
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Fourier Transform of $sgn(t)$

I'm trying to find the Fourier transform of $sgn(t)$ where $$ sgn(t) = \begin{cases} 1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0. \end{cases} $$ By definition, $$ X(\omega) = ...
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How to find the Fourier transform of $\mathbf 1_{[0,2\pi]}(x)\sin(x)$?

How does one find the Fourier transform of $f(x):=\mathbf 1_{[0,2\pi]}(x)\sin(x)$? I have tried to use the definition from my text: \begin{align*} \hat f(\xi) & = \frac{1}{\sqrt{2\pi}} ...
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Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
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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 ...
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Fourier transform in $\mathbb R^3$

I try to show that $$ \int\limits_{R^3} \frac{e^{i\xi x} d\xi}{\xi^2 - k^2 - i0} = e^{ikx} \int\limits_{R^3} \frac{e^{i\xi x}d\xi}{\xi^2 + 2(k + i0\frac{k}{|k|})\xi}, \;\;\; k,x \in \mathbb R^3 $$ ...
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If periodic function has a discontinuity at $x_0$ its Fourier series cannot converge uniformly on any interval containing $x_0$, why?

I'm reading about Fourier series and in one point my book states the following: Suppose $f$ is a periodic function. If $f$ has a discontinuity at $x_0$, the Fourier series of $f$ cannot ...
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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 ...
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Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
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Does a better formula exist for converting quantum state into binary?

First, would like to say that I am not a mathematician so by no stretch of the imagination could I dream of resolving this problem. I have been working with a concept for copying real matter into ...
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Inverse Fourier transform of absolute value of a given Fourier transform

Given that, for some $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, we have the Fourier transform of $f$ given as \begin{equation} F(k):=\int_{-\infty}^\infty ...
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Fourier Transform, Geophysics, Signal Analysis

Geophysics question I don't know if i'm over simplifying it but it says $$ \omega_0 = 2\pi f_0 $$ where $$f_0 = 1/2\pi $$ because that's the frequency of cosine? And that'd just make $\omega_0 = 1$. ...