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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Discrete Fourier Transform actual frequency of pure tone function

Given that a signal s is amplified at $200Hz$ for $10$ seconds, yielding a sequence $s_t$ for $t = 0, 1, ..., 1999$. We have $s_f = \frac{1}{\sqrt{2000}} \sum_{t=0}^{\infty} ...
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7 views

Show the Hankel transform for Gaussian function

We have Gaussian-like function in $k$-domain $$G(k)=2\lambda^2 \sqrt{2K/\pi}e^{-\frac{\lambda^2 k^2}{2\pi}} $$ using the expressions for the Fourier transform ...
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21 views

Condition for uniform convergence of Fourier series

Let $f$ be a Lebesgue summable periodic function on $[-T/2,T/2]$. I read in Kolmogorov-Fomin's (p.414 here) that if $f$ is bounded on a set $E\subset[-T/2,T/2]$ and for any $\varepsilon>0$ there is ...
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Proof the Fourier Transform is Unitary/Not Unitary

I couldn't think of a better title, but could someone provide a proof as to the unitarity of the Fourier Transform from time to angular frequency with the $\frac{1}{\sqrt{2\pi}}$ in front of the ...
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13 views

Fourier analysis on stock price history

What are some ways by which a person might use Fourier Analysis on stock quote price histories? In particular, I want to learn more about the rate at which stock prices oscillate, and come up with ...
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8 views

How to compute multidimensional inverse Fourier transform

everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field: ...
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10 views

Order of distribution of the zeros of the interference function of periodic oscillations?

Given a finite (or an infinite) number of periodic oscillations of different shapes but even functions, along the abscissa $x∈R$, every such periodic oscialltion may cross on some zero points the ...
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11 views

Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
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10 views

A question about time series analysis with fast fourier transform

I have a question about time series analysis with FFT. from here I have understood, to calculate a periodogram of a time series we should do these to steps: Fast Fourier trasnfomation of time ...
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15 views

What are the prerequisites to understand Affine Invariant Fourier Descriptors?

I need to implement Affine Invariant Fourier Descriptors on matlab, the objective is to compare two objects one reference and other transformed by affine transformation for recognition, my problem is ...
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35 views

Apply delta function on Fourier transforms

I would like to filter out a particular frequency $\omega_o$ from the Fourier transform of a function in the form of $\left<\delta(\omega-\omega_o), \mathcal{F}(f)\right>$. If $f(t) ...
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19 views

Quantum Fourier transform $F_N^2$

What is the square of the quantum Fourier transform? I get $1$ for the first entry in the matrix and $0$ for all other entries.
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Mean value of discrete periodic signals

It is clear that a continuous periodic signal always takes at some point $x_m$ the absolute mean value. Then we could define the absolute mean value of the signal as the value that it takes at $x_m$. ...
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22 views

An inverse Fourier transform of Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\tag{1}$$ The functional equation for $\zeta(s)$ is ...
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25 views

On symbol of an improper intergal

In a paper of Ingham with the title "A note on Fourier transforms" (1933) (see http://jlms.oxfordjournals.org/content/s1-9/1/29.extract), he wrote $\int^{\infty} \frac{\epsilon(y)}{y}dy$, and he ...
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the partial derivative of poisson kernel w.r.t. theta tends to 0 as r tend to 1?

the partial derivative of poisson kernel w.r.t. theta tends to 0 as r tend to 1? how to show it ,thanks!!! seem it diverge,because we have a 'n' term after diff.?
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20 views

Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
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19 views

How to determine if someone understands Fourier transforms

I barely understand Fourier transforms. I'm looking for a simple test or Q&A that when used would tell me if someone else understood Fourier transforms. How would I know if the person genuinely ...
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24 views

White noise, how is its definition sensical

White noise is defined as as noise containing all frequencies. Now, consider the inverse fourier transform of white noise, $R$ being the fourier transoform of the noise: $$\int_{-\infty}^\infty R ...
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24 views

Polynomial division/deflation with FFT

There is a need to divide a polynomial $p(x)$ by polynomial $q(x)$, whereas it is known that the remainder will be zero (i.e. the question is about polynomial deflation). A known method is to use the ...
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6 views

Find the spectral density of white noise $\omega$~(0,2)

I am find the spectral density of a white noise given by $\omega$ ~ (0,2). Could you help me to find it? Thank all.
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33 views

Inverse Fourier Transform of $\cos(c\omega t)$ and $\sin(c\omega t)$

I'm just needing a bit of help to understand the derivation of the inverse fourier transform of $\cos(c\omega t)$ and $\sin(c\omega t)$, in deriving D'Alembert's solution to the wave equation. I get ...
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15 views

Fourier Transform Identity?

$f(x) \in \mathbb{R}$ and $g(x) \in \mathbb{R}$ $$\int\int \mathop{dx \, dy} f(x)f(y)g(x-y) = \int dk \, \left| \tilde{f}(k)\right|^2\tilde{V(k)} $$ All integrals are over all space. Is this true? ...
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What is the best estimation for the following?

Let $f$ be 1-periodic and $f\in L_{p}[0,1]$ where $p>1.$ Let $D_{n}$ $n=0,1,2,..$ be the dyadic partition of $[0,1].$ Consider $$ F_{n}(x)=\frac{1}{|I^{n}_{j}|}\int_{I^{n}_{j}}f(t)dt, ...
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Obtaining generating function via Fourier transform

Series coefficient for a function can be obtained via Fourier transform: $$f^{(s)}(0)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^s \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$ ...
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38 views

Fourier transform of exponent?

Mathematica fails to find a Fourier transform of exponent. Yet according to this page $$\mathcal{F}[e^{2\pi iat}]=\delta(t-a)$$ and via substitution, $$\mathcal{F}[e^{at}]=\delta\left(t-\frac ...
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26 views

I need help resolving my problem with DFT

I've been working to understand DFT and my results are not what I would expect. For clarity, I'm using C for T&E and my question isn't C related. My problem is in the DFT and my understanding of ...
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14 views

Radial Fourier Transformation

I'm trying to solve the Fourier Transformation of a radial function, but I'm having trouble computing the integral. This is the function: Let $\vec x \in R^3$ $$f(\vec x)=Ce^{-\frac{r^2}{a^2}} $$ ...
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8 views

Hopf Algebras Arising From Fourier Transforms?

At least for nice functions, a Fourier series is just a Laurent series on a circle (which means just substituting $z = re^{i\theta}$ into a Laurent series), so I can see Fourier analysis on Abelian ...
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27 views

Coefficients of discrete fourier transform

Let's define discrete fourier transform $\hat{x}(k)=\sum_{t=1}^{n}x(t)e^{-i2\pi tk/n}$ where $x(t)=\hat{y}$ and $y:\mathbb{Z}/12\mathbb{Z}\rightarrow \mathbb{C}$ $y(t)=t(-1)^t , 1\leq t\leqslant 12$ ...
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21 views

Momentum Representation vs Position Representation

I have a question involving the representation of operators in momentum representation and position representation. The question is a little long, so I'll do my best to explain it. We are given an ...
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Derivatives of the Fourier transform vanishing on a countable set: construction

Can we construct a time-limited function $f(t)$ whose Fourier transform $F(\omega)$ has the following property: for a given $\omega_0 \in \mathbb{R}\backslash\{0\}$ and $N\in\mathbb{N}$, we have (1) ...
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31 views

Prove that $\prod_{k=0}^{n-1}cos\pi (x+k/n)=\frac{sin\pi n(x+1/2)}{2^{n-1}}$

$$ Prove\:that\: \prod_{k=0}^{n-1}cos\pi (x+k/n)=\frac{sin\pi n(x+1/2)}{2^{n-1}}\\given\:that\\\prod_{k=0}^{n-1}sin\pi (x+k/n)=\frac{sin\pi n(x)}{2^{n-1}}\\and\\sin(\phi+\pi/2)=cos(\phi) $$ From my ...
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Analytic Continuation of Fourier Transform to a Strip in Complex Domain

This is to prove Theorem IX.13 from the Methods of Modern Mathematical Physics (by Reed & Simon). Let $f$ be in $L^2 (\mathbb{R}^n)$. Then $e^{b|x|} f \in L^2(\mathbb{R}^n)$ for all $b<a$, if ...
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31 views

How to plot this function

How to plot this function in WolframAlpha or some other graphing calculator? $f(x) =\left\{\begin{matrix} 1 & -\dfrac{-2\pi}{3} \leq x \leq \dfrac{2\pi}{3}\\ -1 & ...
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37 views

Fourier inversion of an infinitely divisible multivariate gamma measure represented in polar form.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
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Fourier transform option pricing

With regards to option pricing, what is the motivation for using Fourier transform? Is their an alternative to using Fourier transform.
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30 views

Show that the following gradient has the following property

Using the orthonormal matrix $$ R =\begin{pmatrix} \cos(\theta) &\sin(\theta)\\-\sin(\theta)&\cos(\theta) \end{pmatrix} $$ I can define $$ \begin{pmatrix} x'\\y'\end{pmatrix} = ...
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21 views

Find function that gives Fourier transform value of 1

I am trying to find a function $f(a)$ so that the following expression $$ f(a) e^{-\frac{1}{2}\frac{x^2 + y^2}{a^2}} $$ has a Fourier transform equal to 1 as $a \rightarrow 0$. The reason I am doing ...
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33 views

Fourier transform of this oddly worded function

A function is equal to zero outside a unit area square centered at (0,0) and inside a central quarter-unit area square similarly oriented. Elsewhere the function is equal to unity. I am trying to find ...
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22 views

Question from integral with using fourier's integral

Please explain me how to compute this integral: $$ \int_0^\infty \dfrac{\cos(\omega x)+\omega \sin(\omega x)}{1+\omega^2}d\omega$$
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integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
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How to calculate Fourier Transform of logarithmic function?

Given a random variable (RV) $S$ equal to the sum of two mutually independent (RVs) $X_1,X_2$,i.e.$S=X_1+X_2$ and piece-wise probability density functions (PDFs) of $f_{X_1},f_{X_2}$ are as follow: ...
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25 views

How to calculate the integral $\int_{\mathbb R^n} e^{i\lambda |x|^\alpha + ix\cdot\xi} dx$?

I want to calculate the $n$-dimensional Fourier transform of the function $e^{i\lambda |x|^\alpha}$, where $\lambda\in\mathbb R$ and $\alpha \in \mathbb R$, that is, the value of the following ...
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18 views

Solving functional equation $b(x)=\int b(xy)f(y)dy$

I want to prove that given a real-valued smooth function $f$, the set of functions $b$ solving $b(x)=\int_0^{\infty} b(xy)f(y)dy$ is given by linear combinations of $x^{\sigma}$ where $\sigma$ is a ...
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15 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
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24 views

Fourier inversion of $f(k) \Leftrightarrow \lim_{k \to -\infty } f(k)=0$

Let $k \to f(k)$ be a function and define its Fourier transform as $$ \hat{f} (u) = \int_{-\infty}^{\infty} e^{iux} f(x) dx $$ if $\hat{f} (u)$ is integrable we can get back $f$ by doing the inversion ...
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Finding function given its fourier coefficients

Let $f:[-\pi, \pi] \to \mathbb{R}$ be the step function $f(x) = -1$ if $x<0$, $f(x) = 1$ if $x>0$. The Fourier coefficient of $f(x)$ is given as $\widehat{f}(n) = -\frac{2i}{\pi n}$ if $n$ is ...
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23 views

Inverse Fourier transform of complex hyperbolic functions

I'm trying to solve a boundary condition problem and I got the solution in frequency regime: $$f(w)=\frac{\sinh(a|w|)}{b|w|\cosh(c|w|)-iw\sinh(c|w|)}$$ I'm wondering if there's any analytical form ...
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8 views

Finding a discrete Kalman-type process that produces a given Frequency spectrum

Given a power spectral density from f = -1/2 .. 1/2, is it possible to find a 1st order process that produces this series? In other words, x_i+1 = G x_i + W r_i ...