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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Question about computing a Fourier transform of an integral transform related to fractional Brownian motion

I am trying to show an integral transform has a fixed point. Let $H \in (0,1)$ and consider the following integral transform whose kernel is the density of fractional Brownian motion: $$T_H f(x) = ...
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1answer
519 views

Fourier transform solution of three-dimensional wave equation

One of the PDE books I'm studying says that the 3D wave equation can be solved via the Fourier transform, but doesn't give any details. I'd like to try to work the details out for myself, but I'm ...
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3answers
237 views

Schwartz function problem

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz function. Suppose that $\left|\hat{f}(\omega)\right|\leq1$, $\left|\hat{f}(\omega)\right|\leq\left|\omega\right|^{-4}$. Show that: $ ...
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1answer
87 views

Invertible Fourier integral

I am reading a book on Schroedinger's equation and it says that "The relation between $\psi(x, 0)$ and $\phi(p)$ [where the latter is the amplitude in the $\psi(x,t)$ integral] is obtained by ...
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What is Fourier Analysis on Groups and does it have “applications” to physics?

I am trying to be as specific as possible, but I am extremely unclear about this topic (Fourier Analysis on Groups). In Reed-Simon Vol II (Fourier Analysis, Self-Adjointness) there is some ...
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The mathematics of music - why sine waves?

Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal. But what ...
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DFT - Why are the definitions for inverse and forward commonly switched?

Sometimes the forward DFT is defined with a negative exp, sometimes with a positive and occasionally, including the 1/N term. I see this all over the place online. I don't see how the forward and the ...
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204 views

Are there any applications of Fourier series/analysis in General relativity

I'd like to know if there are any applications of Fourier analysis / Fourier series expansion in General relativity ? I mean how Fourier transform has applications in Quantum mechanics.
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1answer
273 views

Asymptotic error of Fourier series partial sum of sawtooth function

In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement: $$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$ where $\{x\}$ denotes the ...
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2answers
108 views

How can I measure the properties of a Point Spread Function?

What quantity or property can I use that describes by how much a point spread function distorts/blurs an image?
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3answers
2k views

Why we need frequency domain?

I am a beginner of Fourier analysis, but my major is economics, I have not much of idea about frequency domain. My question is, since we have whole set of theory to work on time domain, such as time ...
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4answers
596 views

Application to Fourier series

I have seen the following problem in a test, and there are some elementary solutions to it. I am curious if there is a solution involving Fourier series. Here it is: Let $(a_n),(b_n)$ be two ...
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2answers
474 views

Does the vector space spanned by a set of orthogonal basis contains the basis vectors themselves always?

I used to think that in any Vector space the space spanned by a set of orthogonal basis vectors contains the basis vectors themselves. But when I consider the vector space $\mathcal{L}^2(\mathbb{R})$ ...
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0answers
231 views

2 dimensional Fourier transform integral

I'm trying to calculate the two dimensional Fourier integral $$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$$ with $\vec{R}=(x,y)$. Switching to ...
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0answers
68 views

Means to classify data streams or compare similarity

Last year I converted some Matlab code into c to run on embedded Linux. I'm an engineer and normally shy away from maths, but this got me thinking about different ways to classify data or compare the ...
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1answer
538 views

Fourier transform of Fourier coefficients, etc

I have some functions, which are periodic with period 1. Let one of them be $g$. Function $g$ has the following form $(K\rightarrow\infty)$: $$g(x)=\sum_{j=1}^K h\left(\frac{j}{K},x\right) $$ ...
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2answers
147 views

moment of wrapped function (flaw in a proof)

I know the moments of a real-valued function $f$: $$\int_{-\infty}^{\infty}x^{n}f(x)dx=\begin{cases} 1 & ,\ n=0\\ 0 & ,\ n=2,...q-1\\ c & ,\ n=q \end{cases} $$ What can I say about the ...
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2answers
81 views

Should I combine the negative part of the spectrum with the positive one?

When filtering sound I currently analyse only the positive part of the spectrum. From the mathematical point of view, will discarding the negative half of the spectrum impact significantly on my ...
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1answer
272 views

Fourier (Hankel?) transform of a discrete set of radial points (question from a chemist!)

I'm sorry because I'm not a mathematician so that my question may look a little bit messy. I have tabulated values [1] of a 3 dimensional radial function $f(r)$: ...
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1answer
523 views

Exponential Decay of Laplace Coefficients

Laplace coefficients are Fourier coefficients used in Celestial mechanics calculations $$ b^n_s (\alpha) \equiv {1 \over \pi} \int_0^{2\pi} {\cos n \phi \over (1 - 2 \alpha \cos \phi + \alpha^2)^s} d ...
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0answers
102 views

Complex Multiplications Decomposition Into Lifting Schemes

I have tried searching for explanations for breaking down lifting schemes. The context of this question lies in complex multiplication. The authors have decomposed complex multiplications through ...
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1answer
328 views

Fourier transform on a simple smooth 1-manifold

Assume a very simple smooth 1-manifold, with a single chart covering, What I'd like to know is, can we use and Fourier transform for functions on this manifold just as we did for the case of ...
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1answer
404 views

Intuition behind the scaling property of Fourier Transforms

I had a course in PDE last year where we used fourier transforms extensively; I understand the rules of manipulation and can prove the scaling theorem directly from the definition using a ...
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1answer
495 views

Wavelet Theory — where do I start?

I am in the process of implementing a Fixed-Point Fast Fourier Transform. The Fixed-Point FFT requires mathematical background in the area of wavelets and lifting schemes. What are good ...
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1answer
151 views

How large are the second, third, fourth, etc. ringing artifacts in Gibbs phenomenon?

I've read that in the Gibbs phenomenon, partial Fourier series will over- or underestimate a function's value in neighborhoods of jump discontinuities. Specifically, the maximum error will converge to ...
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1answer
457 views

Discrete Fourier Transform Matrix (DFT)

When does it occur that the eigenvalues of a Discrete Fourier Transform matrix are distinct?
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1answer
325 views

($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$

I'd like to find the $n$-dimensional inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$ i.e. $$ \int_{\mathbb{R}^n} \frac{1}{ \| \mathbf{\omega} \|^{2\alpha}} e^{2 \pi i ...
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1answer
199 views

Stuck on complex integral, approximate?

I've been stuck on a particular integral I encountered. I don't need an exact solution, I doubt it even exists. $$f(x)=\frac{e^{-i (r+R-k) x} \left(i-2 e^{i (r+R) x} r x-R x+e^{2 i r x} (R ...
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1answer
4k views

Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ ...
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1answer
204 views

Fourier transform of function in $L^{4/3}$

Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f)$. Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by ...
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1answer
87 views

A naive question on Haar measure and the module of automorphism

people define haar measure to be left invariant,Weil define module of a automorphism to be the quoient of aX and X,where aX denote X changed under operation “a",if it is left invariant,should module ...
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1answer
199 views

A calculation involving the normalized area measure

I am reading about the Dirichlet Space right now. The definition of a Dirichlet space is the set of all holomorphic functions in the unit disc that are finite with respect to the semi-norm: $\mid \mid ...
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81 views

Simplifying imaginary term of jt in Fourier

I can't figure this out. Don't blame me, but please answer this question. I want to simplify this term: $3(e^{5it} + e^{-5it})$ It would be nice to see a detailed workout. I know the answer is ...
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1answer
71 views

Is the co-domain of a Hilbert transform of a function the same as the function itself?

Let $f:\mathcal{D}\to\mathcal{D}$ be a function whose domain and co-domain are $\mathcal{D}$. Let $\hat{f}$ be its Hilbert transform, which is defined as $$\hat{f}(t)=\mathcal{H}(f(t))=\frac{1}{\pi} ...
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261 views

Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc?

Consider any continuous $2\pi$ periodic function (of bounded variation) $f : \mathbb{R} \to \mathbb{R}$ and its fourier series given as $f(\theta) = \frac{a_o}{2} + \sum\limits_{n = 1}^{\infty} ...
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1answer
160 views

reference for Fourier series for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$

I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. ...
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1answer
3k views

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft. Greetings I'm trying to rebuild a signal from the frequency, amplitude, and phase obtained after I do ...
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2answers
7k views

FFT bins from exact frequencies

I'm trying to understand a few concepts about Fourier Transforms (mainly in the context of signal processing). Let's suppose a signal is sampled at 10kHz and that the FFT size is 1000. If 1000 ...
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0answers
96 views

Does the Fourier transform of sequence $f_n\to f$ in $L^2$ converges almost everywhere to $Ff$

$\mathbb K$ is a $\textit{local field}$ if it is any totally disconnected, locally compact, non-discrete, complete field.For examples: $\mathbb Q_p$, any finite extension of $\mathbb Q_p$, field of ...
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1answer
74 views

Terminology concerning Convergence of Fourier Series

Let $f\in L^1(\mathbb{T})$, and $\sum_{n}a_{n}e^{int}$ its Fourier series. Fix a $t_{0}\in \mathbb{T}$. Suppose $\sum_{n}a_{n}e^{int}$ converges at $t_{0}$. But if it is still possible that ...
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1answer
93 views

inequality for an integrable real valued function with a compactly supported fourier transform

Let $f$ be an integrable function on $\mathbb{R}$ where support($\hat{f}$) $\subseteq$ [$-\gamma, \gamma$] for some $ 0 < \gamma < 1$ Prove that | $f(x) - f(0)$| $ \leq c \gamma$ |x| ...
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0answers
234 views

FFT signal post processing

This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole. I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
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1answer
2k views

Convolution & DFTs: How much zero padding is necessary to avoid circular convolution?

When performing discrete [spatial] convolutions in the frequency domain, how much zero-padding is necessary to avoid the effects of circular convolution? I have a book that almost certainly answers ...
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1answer
227 views

When has this sequence a triangular shape?

Can someone explain to me why y[n] has a triangular shape? From what I have found, there is a specific range $n_0$ where y[n] is triangular. But right now I have no clue where to start. $^2$ Search ...
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99 views

Is the following operator a projection?

Let $P$ be a projection defined on $L_2(\mathbb{R})$ by multiplying with the function of value $1$ for $-1<x<1$ and $0$ otherwise. Let $F$ be the Fourier transform and let $F^{-1}$ be its ...
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236 views

Laplace Eigenfunction: Show Eigenvalue is Positive Using Fourier Transform

Problem: Let $ \lambda\in\mathbb{R}, u $ a smooth function, not identically zero, defined on a neighborhood of the unit disc satisfying $ \Delta u+\lambda u = 0 $ in the interior of the unit disc and ...
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1answer
180 views

Cooley Tukey DFT splitting doubt (should be simple)

I can't understand the basic principle on which the Cooley Tukey algorithm is based, the algorithm says I can split in two parts the DFT computation like in the following $$\begin{matrix} X_k= ...
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0answers
129 views

Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
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2answers
1k views

Proof of Fourier Transform

Where F is the fourier transform, how can you show that $$\mathcal F(x\cdot f(x)) = −i \frac{d\mathcal F}{dw}.$$ I understand that you are meant to apply the inverse transform to the left hand side, ...
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139 views

Estimate the Hilbert transform

Let $1\leq p<∞$: Suppose that there exists a constant $C>0$ such that for all $f\in S(\mathbb{R})$ with $L^p$ norm one we have $$\biggl|\{x:|H(f)(x)|>1\}\biggr|\leq C.$$ Here $H(f)$ is ...