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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show $||f_x||_{L_p}=||f||_{L_p}$ for all p

let f be defined on [0,2$\pi$) denote $f_x(y)=f(y-x)$ how do i go about showing $||f_x||_{L_p}=||f||_{L_p}$ for all p i was trying to write out f in fourier series, and use the fact that the ...
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311 views

Calculating the inverse Fourier transform of two given functions

I need to calculate the inverse Fourier transform of the following functions: $\displaystyle f(w) = e^{(-\pmb{i}5w)} * {\rm sinc}(2w) $ $\displaystyle g(w) = ...
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307 views

reference request for proof of Gibbs phenomenon at jump discontinuities

Please suggest a reference for a proof of Gibbs phenomenon at jump discontinuities of a function.
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474 views

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function - a reference request

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to ...
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404 views

Derivation of FFT

Can someone please share a link or source where I can find the derivation of FFT(base-2) from the DFT. I need to put this in latex for my thesis and am finding so many different explanations that I ...
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459 views

Fourier Transforms

I'm having a terrible time trying to understand Fourier transforms. I'm very visual so leaving the $X,Y,Z,t$ domain is not working form me :) I'm trying to figure out the basics at the moment. ...
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788 views

Multiplication using an FFT

I'm exploring the use of FFTs for multiplication, but even with simple examples it seems to go wrong. For example, here I'm trying to multiply $1$ by $2x$ (code is in matlab, but I think you can ...
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84 views

DFT of a certain function

using sum of geometric progression and inverse DFT we can show: $${\sum_{j=0}^{k-1}\frac{\exp(2\pi ij\frac{u+1}{k})}{\exp(2\pi i\frac{j}{k})-a}=k\frac{a^{u(mod\ k)}}{1-a^{k}}}$$ My question is how to ...
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Derivation of Fourier Transform?

So from the fourier series, we can simplify it further and use trig identities to get the following: $$ f(t) = \frac{a_0}{2} + \sum^{\infty}_{n=1} \left(\frac{a_n}{2}+\frac{b_n}{2i}\right)e^{i n ...
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Derivation of Fourier Series?

Can someone point me to the full derivation of the Fourier Series? I'm having problems understanding how the a's and b's coeffients are worked out.
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spectrum and phase of function in frequency domain

This must be a very basic question but I am a finance student just learning some basics about Fourier Transformation to apply to time series analysis. I did a fourier transform on a function in time ...
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283 views

Convolution inequality

Let $u$ and $v$ be two $L^1(\mathbb{R})$ functions such that $\|u\|_{L^1} \le \|v\|_{L^1}$ and $f$ is non-negative $L^1(\mathbb{R})$ with non-negative inverse Fourier transform. Is it true that for ...
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Codomain of the Fourier transform

Among other things, the Fourier transform maps functions from $L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $L^1(\mathbb{R}^n) \to C_0(\mathbb{R}^n)$ (continuous functions vanishing at infinity), and ...
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521 views

Quantum wave packet propagation, how to use it in FFT?

So I used the split step method on the Schrodinger equation and have produced the following equation: $\Psi(x,t+dt)=F^{-1} \left\{ ...
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216 views

Fourier Transform of an unsteady function

I am a little stuck at the moment with the Fourier transform of a function $w(z)$ that is defined piecewise and discontinuous at $z = 0$: $$w\left(z\right) = \begin{cases} -\gamma_1 e^{-\gamma_{1}z} ...
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505 views

Gaussian fixed point Fourier transform

We know that $\text{exp}(-\alpha |x|^2)$ is a fixed point for the unitary Fourier transform if $\text{Re } \alpha > 0$. I know many arguments to show this (contour-integration and ...
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How to perform a Fourier transform in spherical coordinates?

For a function $f(r, \vartheta, \varphi)$ given in spherical coordinates, how can the Fourier transform be calculated best? Possible ideas: express $(r,\vartheta,\varphi)$ in cartesian coordinates, ...
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405 views

What does [n] mean here?

I am reading this document. What is the meaning of $[n]$ ? Is it power set of $\{1,2,3...n\}$?
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Proving a fourier series identity

I studied fourier series as an undergrad and grad. student in EE but did not fully grasp the concepts. Now that I am involved in medical imaging (MRI) understanding the basics of fourier series and ...
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317 views

Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x $

This is a stronger one related to the question Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x $. $F_n(x) : [0,1] \rightarrow \bf R $, for $1 \leq i \leq ...
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Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x $

$f_n(x) : [0,1] \rightarrow \bf R $, and $f_n$ is $\frac{1}{n}$-periodic, $\max f_n(x) = n$, $\min f_n(x) = -n$. As $n$ and $v$ goes to infinity simultaneously, prove the convergence of ...
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Equidistributed sequence and Riemann integrable function

Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence which is equidistributed in $[0,1)$. (a) Is it true that $$\frac{1}{N}\sum_{n=1}^N f(x+\xi_n)$$ converges to ...
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Is this a bounded sequence?

Let $0<a\le 1$ be fixed. Is the sequence $$a_N=\int_1^N x^{-a}e^{ix} dx$$ bounded?
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learning algebra and harmonic analysis

I've revised my question a bit in response to the (very helpful) advice so far-- I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather ...
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672 views

Isoperimetric inequality implies Wirtinger's inequality

Let $C: x=x(t), y=y(t), a\le t\le b$ be a $C^1$ closed curve (not necessarily simple).The isoperimetric inequality says that $$ A\le \frac{\ell^2}{4\pi},$$ where $$A=\left|\int_C y(t)x'(t) ...
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Can you provide help with interpreting this periodogram?

I'm trying to track down the source of some wonky data. The data are response times (RTs) collected from humans using a computer keyboard. Here's a histogram of the RTs, binned to 1ms: Obvious is a ...
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450 views

FFT gives ghost frequency

Imagine the varying frequency generator, which feeds this frequency to the oscilloscope. Oscilloscope is turned to show signal spectra with FFT. The image on the screen of the oscilloscope is as ...
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Fourier cosine transform

Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$. I can solve this question only if there is $x$ instead of $x^2$ in the exponential ...
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509 views

Fourier analysis for waves

I'm studying physics, so I'm sorry if I'll write some inexact things in this post. I wish you can understand me. If we have 1D wave equation: $$\frac{\partial^2 \psi}{\partial ...
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486 views

A sequence not equidistributed in [0,1]

Let $$a_n=\left(\frac{1+\sqrt{5}}{2}\right)^n.$$ For a real number $r$, denote by $\langle r\rangle$ the fractional part of $r$. Why is the sequence $$\langle a_n\rangle$$ not equidistributed in ...
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129 views

A limit related to the Gibbs phenomenon

Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x(N)$ be the number in $0<x<\pi/N$ such that $D_N(x)=1$. Is $$\left|\int_{x(N)}^{\pi/N} D_N(t)\mathrm dt ...
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Getting the value of a Fourier Transform, problem with the complex part

I'm currently trying to do some Fourier transformations, or at least trying to understand them. The only thing I'm worried about is the complex part of the function. All I have is some basic, self ...
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571 views

Fourier series analog of a formula in Fourier transform

Every Fourier transform formula that I know of has a corresponding Fourier series analog, except the multiplication formula $$\int_{-\infty}^\infty f(x)\hat{g}(x) dx=\int_{-\infty}^\infty ...
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What is the Fourier transform of the product of two functions?

Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer. The motivation behind the question is homework, but this is a basic principle in ...
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975 views

Accessible proof of Carleson's $L^2$ theorem

Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$). Some time ago I ...
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Interpretation of Poisson Summation Formula

This question arises from a Fourier transform class I took about a year back. The poisson summation formula is: $$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - ...
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how to calculate fourier transform of a power of radial function

Is there an easy way to see that the $n$-dimensional Fourier transform of $1/|x|^a$ is equal to $1/|x|^{n-a}$ (up to multiplicative constant) where $x$ is an $n$ dimensional vector (assuming that $ 0 ...
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Recommended books/links for Fourier Transform beginners?

I am a student taking engineering course and wish to learn more about Fourier Transforms. It seems very useful. Would highly appreciate it if anyone could advise me where to start.
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Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
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Does rapid decay of Fourier coefficients imply smoothness?

Under the isomorphism of Hilbert spaces $L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n$, smooth functions on the circle are mapped to rapidly decaying sequences (see wikipedia). Is the ...
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1answer
731 views

Pointwise but not uniform convergence of a Fourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
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947 views

Convergence of a Fourier series

Let $f$ be the $2\pi$ periodic function which is the even extension of $$x^{1/n}, 0 \le x \le \pi,$$ where $n \ge 2$. I am looking for a general theorem that implies that the Fourier series of $f$ ...
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How do I compute the eigenfunctions of the Fourier Transform?

In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections ...
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How to sketch a sinc function by hand?

I have to do this for an upcoming exam, but cannot find anywhere (in the textbook or online) how to do this. I only really need to know a couple points to plot it... when x = 0, and then the earliest ...
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Derivatives distribution

Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$ Then how to prove that $f$ is a constant? I had ...
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398 views

Fourier transform of a special Schwartz function

In Classical Fourier Analysis by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ ...
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Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?

If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ ...
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747 views

Rigorous definition of convolution with the unit doublet

The unit doublet is a symbolic object whose convolution with a differentiable function is supposed to give the derivative: $$(x * u_1)(t) = \frac{dx(t)}{dt}$$ See also: ...
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Extracting exact frequencies from FFT output

Say I pass 512 samples into my FFT My microphone spits out data at 10KHz, so this represents 1/20s. (So the lowest frequency FFT would pick up would be 40Hz). The FFT will return an array of 512 ...
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Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...