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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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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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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25 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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23 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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23 views

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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34 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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18 views

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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28 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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20 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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18 views

Fourier Transform Table [on hold]

Does anyone have a good Fourier Transform Table which is synced with: F(w)=1/2pi * integral of f(x)*e^-iwx dx from -infinity to +infinty ? Thanks in advance.
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68 views

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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26 views

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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50 views

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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10 views

What is the significance of laplace and fourier transform [on hold]

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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12 views

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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17 views

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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35 views

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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28 views

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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74 views

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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40 views

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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24 views

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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46 views

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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58 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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23 views

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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19 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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51 views

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 $ ...
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Fourier Series: even extension and Parseval Identity

I'm trying to solve this exercise but I have some problems, because I haven't seen an exercise of this type before. $f(x)= \pi -x$ in $[0, \pi]$ Let's consider the even extension of f(x) in ...
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43 views

Fourier transform of sinc function.

Let us consider the sinc function: \begin{equation} {\rm{sinc}}(x)= \begin{cases} \frac{ \sin(\pi x)}{\pi x} \qquad &x \not= 0,\\ 1\qquad & x=0, \end{cases} \end{equation} ...
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Is the inverse Fourier transform a “linear transform”?

Consider the inverse Fourier Transform and the Fourier Transform: $$f(x) = \int_{-\infty}^\infty F(k)e^{2\pi i k x}dk \\ F(k) = \int_{-\infty}^\infty f(x)e^{-2\pi i k x}dx$$ The Fourier transform ...
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Normaliztion in$L^{p}$ and $L^{q}$

Given a function f in $L^{p}$ and $L^{q}$ where $0<p,q<\infty$ Is f can always be normalized s.t. $\left\Vert f \right\Vert_p=\left\Vert f \right\Vert_q=1$
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Even or odd function. Fourrier coefficients

This is probably a very easy question, but I can't find the answer to it.. I'm working on Fourier coefficients and whether or not the integrals become zero. As far as i'm concerned this integral ...
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Positivity of the Fourier transform of a certain function

I am trying to show that the Fourier transform of $\cosh(x)^{-\nu}$ is positive for every $\nu>1$. I know that such a function has even Fourier transform and... that's about it. Could you suggets ...
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48 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
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63 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
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33 views

Example of Parseval's Theorem

In the textbook "Mathematics for Physics" of Stone and Goldbart the following example for an illustration of Parseval's Theorem is given: Until 2.42 I understand everything but I don't understand ...
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Using FFT to compute DFT of a polynomial

i currently studying about FFT and DFT and we were given simple question: Use the recursive FFT to compute the DFT of this polynomial of '3' degree: $$-1\:+\:4x\:+\:3x^2$$. So, i go to this ...
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Equality with fourier transform

I have problem with the following equality where the Fourier transform appears: Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, ...
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27 views

fft phase plot of pure sine function, why so messy?

I am plotting the phase plot of $sin(2*pi*60*x)$ in the frequency domain. Ideally, we should only see two peaks. How come this is not the case in matlab? ...
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Phase difference of two signal of different frequency

Currently, I have two signals, the main components of both signals are 60Hz, but both also have weaker response at 180Hz + small amount of noise. As shown in the photo below, I want to find the phase ...
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67 views

Questions about the Fourier transform as a unitary transform

As far as I know, the Fourier transform is a (linear) unitary transform: $T: \textbf{L}^2(-\infty, +\infty) \rightarrow \textbf{L}^2(-\infty, +\infty)$ where the basis functions {$e^{i \omega x} | ...
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52 views

Why is the integral from 0 to 1 of $\sin(2\pi nt) \sin(2\pi mt)$ equal to 0,5 if $m$ and $n$ are equal?

I am interested in this result because I am studing Fourier Series. By the way, although I have studied Mathematical Analysis, my background is not so good. Could you please explain why the integral ...