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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Taking the Fourier Transform of a specific function.

I have this function: $$f(x) = \prod_{p\text{ is prime}} \left(1 - \frac{x^2}{p^2}\right)$$ Now, this function can be said to be an infinite degree polynomial with zeros on each of the primes and ...
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Frequency response of a linear, shift-variant system

I am working my way through recorded lectures and a textbook related to DSP, and have come across a question that I am not sure how to answer. This is probably just due to how new I am to these ...
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45 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
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Absolute square in deriving Fourier transform variance

I'm having some trouble understanding how to derive the variance of the Fourier transform. This is for an image, i.e., it's a 2D transform. The variance is $|\hat{I}(\xi,\eta)|^2$, the absolute ...
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Why fourier spectrum of a signal repeats at every multiple of sampling frequency?

Discrete Fourier transform of a signal is expressed as $$X[k]=\frac{1}{N}\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}$$ which can be expressed as: $$X[k]=\frac{1}{N}\sum_{n=0}^{N-1}(\cos(-2\pi kn/N) + ...
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23 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
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Fourier spectrum reflected across origin and Nyquist frequency

Recently I've been trying to figure out what's the point of negative frequencies produced by the fourier transform. One answer was it's just there to make calculations more elegant. It could be ...
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Generalization of the Jacobi-Anger expansion to higher harmonics

I know the Jacobi-Anger expansion relation which gives the Fourier development of $e^{i z \cos(\theta)}$ and ${ e^{i z \sin(\theta)} }$, such that $$ \begin{cases} e^{i z \cos(\theta)} = ...
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28 views

If $u : \Bbb R \to \Bbb R$ satisfies $u' + 2\pi x u = 0$, why does $\hat{u}$ (the Fourier transform) also satisfy this?

I'm trying to understand why if a function $u : \Bbb R \to \Bbb R$ satisfies the differential equation $u' + 2\pi x u = 0$, then so does the Fourier transform. The properties I have that I can use ...
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32 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
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Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
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Analog of the Fourier transform on a bounded domain?

Suppose $\Omega \subset \mathbb{R}^d$ is a bounded, simply-connected domain with whatever other "niceness" properties are necessary for this question to make sense. Define the "Fourier transform" over ...
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16 views

Question about a proof of Riemann localization theorem

The Riemann Localization Theorem states that Let $f \in L_{2 \pi}^2$ and $x_0 \in \mathbb R$. Then $$ \lim_{n \to \infty} (S_nf)(x_0) = f(x_0)$$ if and only if there is a $\delta \in (0, \pi)$ ...
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Sum of unitary complex numbers

Let us define: $$\varphi(x,n,t):=\frac{1}{n}\sum_{y=1}^n \left| \sum_{k=1}^n e^{2ik\pi (x-y)/n + 2i \sin(2k\pi/n) t} \right|$$ Does somebody have an idea how to prove that $$ \sup_{x=1,...,n} ...
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28 views

integration and convolution

Please can some one help me on the following integration. $$ G(\nu)=\frac{1}{\Delta t}\int_{t_a - \frac{\Delta t}{2}}^{t_a + \frac{\Delta t}{2}} f(t_a -t)e^{-2\pi\nu it}dt $$ where ...
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26 views

Does there exist a non-negative valued compactly supported function such that its Fourier transform only vanishes at a given point?

My question is as follows: Given $t_0\in\mathbb{R}$. Does there exist a non-negative valued compactly supported function $f\in L^1(\mathbb{R})$ such that its Fourier transform, $\widehat f\left( t ...
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Integral involving rapidly decreasing functions [on hold]

Let $F,\varphi\in S(\mathbb{R}^n)$, the Schwartz space of rapidly decreasing functions. Is this enough to guarantee that the integral $$\int_{\mathbb{R}^n} F(x)\varphi(x)dx$$ is well-defined? Why or ...
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27 views

How to evaluate this (Fourier) integral? [duplicate]

Does somebody know how to evaluate $$\int_{\mathbb{R}^n}\frac{e^{i\langle\xi,x\rangle}}{\|\xi\|_2^2}d\xi$$ for some given $x\in\mathbb{R}^n$ and $n\in\{1,2,3\}$?
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29 views

Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
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25 views

Check my Proof: Possible kernels for an integral transformation

Edit: Meanwhile I found a solution by myself, see below. Feel free to comment. Playing with some problem in quantum physics, I arrived at the following integral equation $$\frac{1}{2\pi}\int dk ...
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Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
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Calculation of Fourier Transform

I am trying to calculate the Fourier Transform (with respect to k) of the following function: $\frac{1}{\sqrt{k^2 + 1} - C}$ where $C$ is some complex number. Does anyone knows how to do that? ...
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36 views

Find Fourier Coefficients

I am asked to find the coefficients for $f(t)=\sin^{2}(5t)$ $$Period =\frac{\pi}{5}$$ so I wrote $$a_n\cdot\sin(\frac{n\pi{t}}{\frac{\pi}{10}})=\sin^{2}(5t)$$ $$a_n\cdot\sin(10n{t})=\sin^{2}(5t)$$ ...
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Scaling for Matlab fft operation?

I have a $N$ complex signal samples (QPSK) and I am creating an OFDM signal. When I am doing a IFFT operation in matlab, I use following command: $$Y=(dft/sqrt(N))*ifft(X),$$ where $X$ is the input ...
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An upper bound for the Fourier coefficient of the “infinite cake” function

Consider a function $x_{s_n} (t) = s_n$ for $t\in[-\frac{s_n}{T_0}, \frac{s_n}{T_0}]$ and $x_{s_n} (t) = 0$ for $t$ everywhere else, with period $2T_0$. Now let $s_n=\frac{1}{n^2}$, and define the ...
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31 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
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21 views

Computing the accumulated power of an image's Fourier transform

I need to compute the accumulated power of an image. The purpose is to verify the 1/$f$ power law in natural images. I'm not sure how to do this. I've done a fast Fourier transform of the image and ...
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Does these sequence and series converge?

Let $f\in C^1[-\pi,\pi]$ st $f(-\pi)=f(\pi)$ and define $$a_n=\int^{\pi}_{-\pi} f(t)\cos nt dt\,$$ for $n \in\Bbb{N}$ . Then does the sequence $\{na_n\}$ converges? And does the series ...
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Find Coefficients from already fourier function

Hello I have this function and I'm asked 1.Find the period for $f(t)$ 2.Find the coefficients $a_n$ and $b_n$ $$f(t)=2(cos(2t+\frac{\pi}{4})-sin(6t-\frac{\pi}{2}))$$ I know that the period for ...
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Fourier series of a periodic odd function

Given $\ f(\theta)=\theta(\pi-\theta)$ is a $2\pi$-periodic odd function on $[0,\pi]$. Compute the Fourier coefficients of $f$, and show that $\ f(\theta)=\frac{8}{\pi} \sum_{\text{$k$ odd} \ \geq 1} ...
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Fourier transform: no time localization but an inverse exists. How can these properties go together?

Take for example one period of a sine: $f(x) = \{\sin(\omega_1x) \; \mathrm{if} \; x \in [0, 2\pi) \;; \quad 0 \; \mathrm{elsewhere} \}$ If we now translate $f$, then according to the argument that ...
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Fourier Transform of $f(x) = \exp(-\pi ax^{2} + 2\pi ibx)$

I was trying to take the FT of $$f(x) = \exp(-\pi ax^{2} + 2\pi ibx)$$ This is just the shifting rule applied to the FT of $$g(x) = \exp(-\pi ax^{2})$$ which is given by $$\hat g(k) = ...
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What is the significance of laplace and fourier transform [closed]

I know what laplace and fourier transforms are used for but i want to know if these operators show some properties or are they just mathematical operators to simplify our work
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2D Fourier transform of $x.\frac{\partial f}{\partial x} (x,y)$

What is the simple form of the 2D fourier transform of the following functions: $$ g(x,y)=x.\frac{\partial f}{\partial x} (x,y) $$ And $$ h(x,y)=x.\frac{\partial f}{\partial y} (x,y) $$
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42 views

How is the exponential in the Fourier transform pulled out of the integrand?

I'm looking at Fourier Transforms in a Quantum Physics sense, and it's useful to associate the Fourier Series with the Dirac Delta. The book I'm using follows this argument (Shankar, Quantum ...
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Harmonics conditions for a plucked string

Given a plucked string which is taken on the interval $[0,\pi]$, and it satisfies the wave equation with $c=1$. The initial position of the string is: $\ f(x) = \frac{xh}{p}$ ($0\leq x\leq p$), and $\ ...
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Showing $\sum_{n=-\infty}^{\infty}\exp\left(-\pi an^2+2\pi ibn\right)=a^{-\frac{1}{2}}\sum_{m=-\infty}^{\infty}\exp\left(-\frac{\pi(m-b)^2}{a}\right)$

How do I show that \begin{align} \sum_{n=-\infty}^{\infty} \exp\left(-\pi a n^2 + 2 \pi i bn\right) = a^{-\frac{1}{2}} \sum_{m=-\infty}^{\infty} \exp\left(-\frac{\pi(m-b)^2}{a}\right) \end{align} is ...
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What is the relation between analytical Fourier transform and DFT?

First of all let me state that I searched for this topic before asking. My question is as follows we have the Analytical Fourier Transform represented with an ...
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Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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$f\in L^1\cap L^2$ implies $\hat f \in L^1$?

Given $f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$. The Riemann-Lebesgue lemma and the unitarity of the Fourier transform on $L^2$ implies that $\hat f \in L^2\cap C_0$ where $C_0$ are continuous ...
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strange transform of dirac delta function

one of our homework solutions states that $$\delta(x)\equiv\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega e^{-i\omega x} $$ is the Fourier transform of the dirac delta function. But according to the ...
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When has the Fourier transform for some values equal values?

Definition We take a function $F : \mathbb T^n \rightarrow \mathbb R$ that is even ( $F(x)=F(-x)$) and continuous (hence bounded), where $\mathbb T^n$ is the $n$-dimensional Torus. Now we define the ...
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Inverse Fourier transform of $F(\omega)=\frac{1}{\sum_{k=1}^N e^{i \omega z_k}}$

I am looking for the inverse Fourier transform of \begin{align*} F(\omega)=\frac{1}{\sum_{k=1}^N e^{i \omega z_k}} \end{align*} where $z_k \in Z$. But I don't know how to approach it. This reminds ...
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59 views

On the proof of Fejér-Riesz theorem

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf First of all, I didn't ...
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Periodic functions proof

I need some help here. Let $f$ be a $2\pi$-periodic function, and define for an arbitrary $k\in\mathbb N$ a function $g(x) = f(kx)$. Show that $g$ is also $2\pi$-periodic. What I've done: $$ g(x) ...
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Give a suitable way to study Fourier Transforms:

Give a suitable way to study Fourier Transforms. In the website called the fourier transform, gives somewhat good approach to meet it. But, I need to clarify onething. I am doing my pure papers ...
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Fourier transform of delta function $\delta(x)$ and my defined function $s_1(x)$ and $s_2(x)$

$\delta(x)$ and $s_1(x)$ are $0$ if $x\not=0$, if $x=0$, then $\delta(x)=+\infty$ and $s_1(x)=1$, respectively. $s_2(x)=1$ if $1\geq x\geq -1$, otherwise $s_2(x)=0$. What is the Fourier transform of ...
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Discrete Time Fourier Transform of the signal represented by $x[n] = n^2 a^n u[n]$

I have a homework problem that I am just not sure where to start with. I have to take the Discrete Time Fourier Transform of a signal represented by: $$x[n] = n^2 a^n u[n]$$ given that $|a| < ...
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20 views

Parseval's Identity, problem with $|a_n|^2$

I'm trying to obtain the Fourier Series of this function: $$f(x)=\begin{cases} \pi -x, x\in [0, \pi]\\ \pi+x, x \in [-\pi, 0) \end{cases}$$ It is a even function, so I can write: ...
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Fourier Transform of a Temperate Distribution

Let $f$ be a temperate distribution. Suppose that $f$ is a solution to the equation $ f'-f= \delta_0 +1 $. I want to find $ \hat{f}$... Here's what I did: Since $ f'-f= \delta_0 +1 $, then $ ...