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 - epicycles.

I have $f(x) = \sin(x)$. I thought that when I do Fourier Transform and construct epicycles, than those epicycles will draw that $\sin(x)$ function (but this is probably not case with $\sin(x)$, cause ...
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63 views

Fourier transform of a potential

I need help computing the distributional inverse Fourier transform of the function $1/|x|^2$ in dimension two. The integral makes sense written as \begin{align} 1/2\pi \int_{\mathbb{R}^2} e^{ix\xi} ...
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395 views

Fourier Transformation: an Animated GIF

Here I found the animated GIF below. I don't get it! Would someone explain it please?
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49 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
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78 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
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186 views

Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
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1answer
66 views

How to solve this heat equation with fourier method

Solve this via Fourier method: $$u_t-u_{xx}=0 \quad\quad 0< x<\pi, \quad t >0, $$ $$u(0,t)=u_x(\pi,t)=0, \quad\quad t \ge 0$$ $$u(x,0)=2\sin\left(\frac{3x}{2}\right) \, ...
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55 views

What is Fourier Space

I know a some basics stuff regarding Fourier Analysis (Fourier series and Fourier transforms), but I've seen the term "Fourier Space" come up and I'm having trouble finding a definition for what this ...
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97 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
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109 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
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21 views

How to find the Fourier components

I've got a sinusodial signal: \begin{equation} \Delta I=\cos(\Delta \omega t + \varphi)). \end{equation} and I would like to rewrite it as the sum of a cosine and a sine signal(without a phaseterm): ...
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96 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...
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56 views

Fourier and $Z$ transform of a signal?

We have $$X(k)=4[u(k-2)-u(k)* d(k-3)]$$ I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra. First of all I think that I need to convert the $u(k)$ and ...
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135 views

Convolution of ring Delta function

Assume $f(r)=\delta(r-R)$ where $\delta(\cdot)$ is a ring delta function. In other word, $f$ is a circular delta function on a circle with radius $R$. I want to do the convolution of $f$ with itself ...
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55 views

convolution of lorentzian with cosine

Hi: I'm reading a text "Fourier Transforms for Pedestrians" and it's a nice text but it skips steps that I sometimes don't understand. The current example that I don't follow is one where the ...
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1answer
94 views

Units of frequency for Fourier transforms

I'm trying to get to grips with the relationship between a signal $x(t)$, its Fourier transform $X(F)$ and the graph representation of $t$ plotted against $|X(F)|$. I've been using Matlab to perform ...
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1answer
135 views

How do we determine the duration of a fundamental frequency using the DFT (or FFT)?

I'm still in the process of learning the details of the DFT (and FFT) and I've just made a test .wav file in Audacity by joining 3 one-second sine waves together. .wav file 1 = 440 Hz, sample rate ...
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29 views

Mean fourier coefficients of a $2\pi$-periodic function are just the usual Fourier coefficients.

Let $f$ be a continuous $2\pi$-periodic function on $\mathbb{R}$. I'm trying to show that \begin{align}\tag{1} \lim_{T \rightarrow \infty} \frac{1}{2T}\int_{-T}^{T} f(x)e^{-ix\xi}dx = ...
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30 views

Fourier transform causing problems in proof for Parsevals formula

$f^{*}$ is the complex conjugate and $\tilde{f}$ is the fourier transform of Ff$. If $g(-\xi)=f(\xi)^{*}$ how does this imply $\tilde{g}(k)=\tilde{f}(k)^{*}$. This result just does not seem true by ...
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95 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
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2k views

MATLAB: Plotting the inverse Fourier transform of a rectangular pulse.

I'm trying to repeat the results in the image below without using the rectangularPulse(a,b,z) function: Here is my unsuccessful attempt: Here is the code I used to create the first image: ...
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81 views

How did Fourier arrive at the following regarding his series and coefficients?

I am reading Karen Saxe's "Beginning Functional Analysis." Perhaps it is poor exposition on her part, but she states: ...Fourier begins with an arbitrary function $f$ on the interval from $-\pi$ ...
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150 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
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95 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
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1answer
52 views

If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?

Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad ...
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131 views

Fourier series with respect to orthonormal sequence

Let $H$ be the space of piecewise continuous $2 \pi$-periodic functions on the real line. For $f$ and $g$ in $H$, consider the inner product $<f,g>=\frac{1}{2\pi}\int_{- \pi}^{\pi}f(x)\overline ...
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24 views

Fourier transform time and frequency question?

So there is an example in my book where g(k) is converted to G(f) and its written $$g(k)\Longleftrightarrow G(f)$$ So: $$a^ku(k)\Longleftrightarrow \frac{1}{1-ae^{-j2\pi f}}$$ My question is, how ...
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79 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
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283 views

Prove $\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$ using f(x)=1-|x| and Poisson summation formula

I'd like to prove $$\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$$ by using the Poisson summation formula. There is a way to do it by firstly taking the Fourier ...
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209 views

Convolution of a sum of shifted delta functions

In the lecture notes for Fourier Transforms and it's Applications on page 212 by Bracewell he talks about representing a signal as a sum of distributions evenly spaced out by a distance p. ...
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46 views

Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples

I am studying the discrete Fourier transform, and in its most basic definition it is an invertible linear transformation on the complex numbers. From Wikipedia: The sequence of $N$ complex numbers ...
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50 views

Can't extract odd function with FFT

I can't correctly extract spectrum from data points of odd function (e.g. $\cos\left(\frac23\pi x\right)$, $16$-points vector $[1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1]$), instead of one function I get a ...
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70 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
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184 views

Meaning of multiplication by $\sin$ in $\omega$-domain

Multiplying some signal, a function of time, $m(t)$ by a cosine $\cos{\omega' t}$ causes a shift in frequency of $m(t)$, by $\pm\omega'$. But what about multiplication by a sine wave, such ...
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24 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
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71 views

Inverse Fourier Transform

I need help solving the following Fourier transform question. Given, $$ X_s(f) = \frac{1}{\Delta T} \sum_{n = -\infty}^{\infty} X\left(f - \frac{n}{\Delta T} \right) $$ $$ H(f) = \begin{cases} 1 ...
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100 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
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30 views

Fourier transform of $|u|^2*u$

Given the Fourier transform of $u$ is $\widehat{u}$, then what can I say about the Fourier transform of $|u|^2u$? Can I represent it by $\widehat{u}$?
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42 views

intuition behind inverse transform of $ cos(\omega_{0} t)$

Hi: After looking around the internet and looking at solutions to similar questions, I was finally able to convince myself of the following mathematically. If $G(\omega_{0}) = cos(\omega_{0} t)$, ...
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1answer
35 views

Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
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179 views

Easiest way to prove integral of $e^{ikx}$ is $\delta(k)$

What is the easiest way to to derive the following equation: $$\int_{-\infty}^{\infty}e^{ikx}dx = 2\pi\delta(k)$$ I understand the equation can be derived by assuming the Fourier integral theorem, ...
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149 views

trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points $X_k$. With N/2 the Nyquist frequency and $X_0$ the DC value. Using the inverse we can then get back the ...
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159 views

Characteristic Function Inversion

I am studying the relationship / bijection between characteristic functions and CDFs. In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function ...
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1answer
33 views

Can one express $f'(x)$ with the same basis as one uses for $f$?

If I have an orthonormal basis $\{\phi_n\}_1^\infty$ in space $L^2(a,b)$ and the generalized Fourier series expansion for $f$ would be: $$f= \sum \langle f, \phi_n\rangle\phi_n,$$ then can one use ...
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50 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
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496 views

Convolution of L1 & L2 function: definition

A book that I'm reading makes the following statement that I'm not sure how to understand: On $\mathbb R^n$, if $f\in L1$ and $g\in L2$, we have: $$\widehat{f*g}=\hat f \hat g$$ How do I read it? I ...
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41 views

Proving that $\langle f, g\rangle = \sum_n \langle f, \phi_n \rangle \overline{\langle g, \phi_n \rangle}$

I have the following problem to solve: If the set of functions $\{\phi_n \}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ and the functions $f, g \in L^2(a,b)$, then show that: ...
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74 views

What are the steps to derive the following inverse Fourier transformations

I'm reading a text which is an introductory text on Fourier transforms. The author has two expressions: $$ F(\omega_{o}) = \frac{1}{\sigma \sqrt {2 \pi} } e^{\Large- ...
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148 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
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If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...