Tagged Questions

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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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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0answers
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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2answers
91 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
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1answer
26 views

Difference between $X(\omega)$ and $X(j\omega)$ notation of Fourier transform

In many reference materials i have come across fourier transform of a function $x(t)$ referred as $X(\omega)$ and $X(j\omega)$. But what is the difference between both the representations. Are they ...
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1answer
30 views

Spectrum of Rank 1 Operators

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and ...
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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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1answer
35 views

Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
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10 views

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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1answer
13 views

Fourier transform $\mathcal{F}_y\left\{\frac{e^{-|y|x}}{y}\right\}$

I have shown that $$\mathcal{F}_y\left\{\frac{e^{-|y|}}{y}\right\} = -i \sqrt{\dfrac{2}{\pi}} \tan ^{-1} (k)$$ where $k$ is the frequency variable. I need to find, however, ...
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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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9 views

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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1answer
40 views

Fourier parseval prove for misunderstanding of second negative exponent sign

See the picture below: I know if the sign is not '-', the following derivation can not continue,but I really want to know why $$e^{itx}\cdot e^{i\tau x}=e^{i(t-\tau)x}$$ How it can be that? I ...
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1answer
16 views

Is diagonal of sum matrix of rank one matrix and circulant matrix.

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in $\mathbb{R}^n, n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
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1answer
17 views

Computing integrals of inverse Fourier transform of function with compact support

Suppose $f \in C^{\infty}(\mathbb{R})$ with compact support, $f(0) = 1$ and derivatives satisfying $f^{(n)}(0) = 0$ for all $n = 1,2, \dots$. Consider \begin{align*} K(u) = \frac{1}{2 \pi} \int_{- ...
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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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1answer
30 views

Expression for the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2}$ [duplicate]

I'm having troubles with the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2} \in L^2(\mathbb{R}^{n})$. For the case $n=1$ I got $\hat{f}(\xi) = \pi e^{-2\pi |\xi|}$ using residues. Does the general ...
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1answer
17 views

Showing $\cos{\pi \frac{mx}{R}}\sin{ \pi\frac{nx}{R}}$ are orthogonal in $L^{2}([0,R])$?

I'm trying to show that $\sin \left(\frac{n\pi}{R}x\right)$ and $\cos \left(\frac{m\pi}{R}x\right)$ are orthogonal using the trigonometric identity that $2\cos{mx}\sin{nx} = \sin((m+n)x) + ...
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1answer
41 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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9 views

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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2answers
62 views

Prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous

How does one prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous? I'm unsure of where to begin. Pertinent definitions: ...
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5answers
37 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
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0answers
80 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
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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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1answer
31 views

Fourier Sine Series and Cosine Series

This is the Fourier Series representation for a periodic function with period 2p, given in my lecture note. $\dfrac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(\dfrac{n\pi t}{p})+b_nsin(\dfrac{n\pi ...
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0answers
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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1answer
23 views

Prove the following relation between Fourier transform pairs

Lets say I have a function $f(x,y)$ and its corresponding Fourier transform $F(u,v)$. I'm trying to show that $f(x+by,dx + y) \leftrightarrow ...
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1answer
25 views

Transforming a function in the dirac delta function

I need help finding $k(\sigma)$ such that the family of functions $$ \delta_\sigma (x,y) = k(\sigma) e^{-\frac{1}{2}\frac{x^2 + y^2}{\sigma^2}} $$ defines the unit impulse $\delta(x,y)$ as $\sigma ...
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0answers
31 views

Is there a continuous and satisfying a certain condition but not integrable function?

Is there a continuous function $f$ on the interval $[0,1]$ which satisfies $$ |f(x+h)+f(x-h)-2f(x)|\leq \mathrm{const} \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta ...
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1answer
19 views

Seperable functions have seperable Fourier transforms

How do I show that if $f(x,y)$ is separable into a product of a function of $x$ and a function of $y$, its Fourier transform $F(u,v)$ is also separable into a function of $u$ and a function of $v$?
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1answer
30 views

Fourier transform of following equation

Say I have the following equation $$ f(x,y) = \left\{ \begin{array}{lr} 1 & \text{if} \;|x|,|y| \leq 1 \\ 0 & \text{otherwise} \end{array} \right. $$ What is the ...
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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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1answer
121 views

Expand $f(x) = x$ in a cosines serie

The task is: Expand $f(x) = x$ in a cosines serie. My approach was to use the trigonometry fourier formula $$ c_0 + \sum_{k=1}^{\infty } a_k \cos (k \Omega t) + b_k \sin (k \Omega t) $$ set $b_k = ...
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0answers
53 views

Fourier Cosine Series question

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &2 \leq ...
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1answer
46 views

Convolution theorem - proof

I'm trying to understand a proof of convolution theorem given here: http://www-structmed.cimr.cam.ac.uk/Course/Convolution/convolution.html In section named "Proof of second statement of convolution ...
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1answer
28 views

Is fourier transform or Wavelet transform better for this applicaiton?

I am currently designing an alogirthm that is either based on Fourier Transform approach, or the Wavelet Transform Approach, or the combination of the two. Since Wavelet is new to me, I am having ...
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1answer
19 views

Coefficient in defining Fourier Transform

I don't know why we define below coefficient in Fourier transform? thanks for your help
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0answers
33 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
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1answer
24 views

Centroid from Fourier Transform

I have a discrete fourier transform from an black and white image and would like to have a rough estimation of the centroid of an white shape in it. As far as I can tell from ...
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1answer
30 views

Can the sum of infinitely many zero amplitude sinusoids converge to any function?

I've read this in a post here (can't remember which - might even have been a comment) that I thought that was the most ridiculous thing I have ever heard. Can someone illustrate mathematically that ...
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0answers
18 views

Showing equality between integral and shifted Fourier series

Let $f\in E[-\pi,\pi]$ and let $f\sim \frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx +b_n\sin nx$ be the fourier series of f in $[-\pi,\pi]$.show that $$\forall -\pi\le c,x\le \pi \quad ...
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0answers
19 views

Solve $\int\limits_{\mathbb{R}^n} e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$

How to calculate $$\int_{\mathbb{R}^n}e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$$ where $\langle \cdot, \cdot \rangle$ denotes the canonical inner product in $\mathbb{R}^n$. I'm trying use ...
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0answers
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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37 views

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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22 views

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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1answer
32 views

Finding $\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$ using Plancherel’s theorem

Suppose $G(\mathbb R)\ni f(x),\mathcal{F}[f](\omega)=\frac{1}{1+|w|^3}$ find the value of $$\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$$ I thought using Plancherel’s theorem \begin{align} ...
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2answers
162 views

2D Fourier transform of $1/(x^2-y^2+q)$

How can I calculate the following 2D Fourier integral: $$ \iint \frac{{\rm e}^{{\rm i}(ax+by)}}{x^2-y^2+q} {\rm d}x\,{\rm d}y, $$ where $q$ is a complex number? If there was a "+" sign in the ...
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3answers
57 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
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1answer
73 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
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1answer
36 views

Integrals using Parsevals Theorem

I've been assigned two integrals to calculate in Fourier Analysis: $$\int_{-\infty}^{\infty}\left(\frac{\sin x}{x}\right)^2dx$$ ...