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 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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89 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 ...
2
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1answer
222 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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3answers
83 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
76 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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315 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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174 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
4k 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
10k 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
99 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
77 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 ...
3
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1answer
106 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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266 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
230 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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101 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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271 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 ...
0
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1answer
182 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= ...
2
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0answers
131 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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3answers
188 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
3
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1answer
258 views

Proving that the characters of an infinite Abelian group is a basis for the space of functions from the group to $\mathbb{C}$

Let $A$ be an arbitrary (possibly infinite) Abelian group. A character $\chi$ is a group homomorphism from $A$ to the multiplicative group of complex numbers. I can prove that if $A$ is finite then ...
2
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1answer
439 views

What is the generalization of Parseval's theorem into spherical coordinates?

what is the relationship between the total power of a function given in spherical coordinates in the Fourier domain: $E_k=\int_{\mathbb{R}^3}|F(k,\Theta,\Phi)|^2k^2 \sin(\Theta)\,dk\,d\Theta\, ...
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1answer
1k views

Using the Fourier integral theorem to evaluate the improper integrals

I'm trying to brush up with Fourier series with Apostol's Mathematical Analysis. I was looking through the Fourier chapter and its Fourier integral theorem. I'm slightly confused on how to approach it ...
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741 views

Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
3
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586 views

How do I find the inverse Fourier transform of a function that is separable into a radial and an angular part?

I need to take the inverse Fourier transform of a function that is initially specified in spherical coordinates: $f(r, \theta, \phi) = \int_{R^3}F(k, ...
3
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1answer
379 views

How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk$, where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!
2
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1answer
582 views

Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where ...
8
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1answer
593 views

Fourier transform of Schrödinger kernel: how to compute it?

Let $$K_t(x)=\frac{1}{(4 \pi i t)^{\frac{n}{2}}}e^{i \frac{\lvert x \rvert^2}{4t}}\quad x \in \mathbb{R}^n,\ t \in \mathbb{R},\ t\ne 0.$$ Clearly this is not a $L^1$ or $L^2$ function with respect ...
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564 views

Fourier transform of vector-valued functions (e.g. differential forms)

Consider $L^2(\mathbb R^n, \mathbb R^m)$. There should be a Fourier transform for these functions, like in the case $L^2( \mathbb R^n, \mathbb R )$. I wonder how these can be defined. The application ...
3
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1answer
959 views

How do I find the Fourier transform of a function that is separable into a radial and an angular part?

how do I find the Fourier transform of a function that is separable into a radial and an angular part: $f(r, \theta, \phi)=R(r)A(\theta, \phi)$ ? Thanks in advance for any answers!
0
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1answer
775 views

LTI: How to derive the impulse response of this system?

Well, i transform g and x into the frequency domain. u[n] = 1, n ≥ 0 u[n] = 0, n < 0 \begin{aligned} x[n] & = u[n] \\ h_1[n] & = (\frac{1}{2})^n u[n] \\ g[n] & = (\frac{1}{2})^n u[n] \\ ...
3
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1answer
187 views

Trouble deriving DE for fourier transform from DE of function

I am trying to derive an equation which is a standard result in physics (the momentum space Schrödinger equation). (Background: The wavefunction is a complex valued function of position coordinates ...
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2answers
1k views

Explicitly reconstructing a function from its moments

Let $f$ be an integrable real valued function defined on $[0,\infty)$. Let $$m_n=\int_0^\infty f(x)x^n \mathrm dx$$ be the $n^{th}$ moment, and suppose that all of these integrals converge ...
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1answer
4k views

Fourier basis functions

What are fourier basis functions? And how do I prove that fourier basis functions are orthonormal?
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1answer
74 views

Help with fourier transforms

I am going through a book and having trouble with reproducing some results mentioned. The aim is to solve for $D_{s}$ from equation (1) below $\int ...
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2answers
4k views

Why is 8x8 matrix chosen for Discrete Cosine Transform?

In JPEG and MPEG, why is 8x8 matrix chosen for Discrete Cosine Transform? Why not any other, say 64x64?
2
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2answers
307 views

Calculating $\displaystyle{\int_0^\infty e^{-i\omega t}dt}$

I was studying Fourier Transform; I could answer to this $$\int_{-\infty}^\infty e^{-i\omega t}dt$$ by Fourier Transform, but I have problem in $$\int_0^\infty e^{-i\omega t}dt.$$ I would be grateful ...
3
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3answers
624 views

What is the relationship between different definitions of Fourier transform?

I always see various definitions of Fourier transform. A standard form is: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ and its attached inversion is ...
3
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1answer
389 views

Extension of Plancherel theorem to $\ell^2$

Can Plancherel's theorem, which was originally defined for $L^2$ spaces (rather, functions in $L^1\cap L^2$) be extended to $\ell^2$ spaces? How would one do that, or is it very obvious/intuitive? If ...
4
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1answer
380 views

Is it possible to for the Cesàro means to converge but the Fourier series to diverge?

I was just wondering if it is possible for the Cesàro means to converge but the Fourier series to diverge. In class we learned that if the Fourier series were to converge, then this limit must equal ...
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1answer
734 views

How can I increase/decrease (frequency/pitch) and phase using fft/ifft

How can I increase/decrease (frequency/pitch) and phase using fft/ifft I think I have the basic code but I’m not sure what to do next PS: It's done in Octave/matlab code Example I have a signal that ...
3
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1answer
372 views

One more question about decay of Fourier coefficients

Let $$f=\sum f_{s}\exp(2\pi isx)\in C^{(p-1)}[0,1]$$ and $$f^{(p)}\ in\ L_2[0,1]\ \ ( \sum\left|f_{s}\right|^{2}j^{2p}<\infty )$$ Does it imply that $f_s=O(s^{-(p+\psi)})$ for some ...
2
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1answer
202 views

FFT and changing frequency and vectorizing FOR loop

I can increase and decrease the frequency of a signal using the combination of fft and a Fourier series expansion FOR loop in the code below but if the signal/array is to large it becomes extremely ...
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3answers
19k views

What is the difference between the Discrete Fourier Transform and the Fast Fourier Transform?

can anybody answer this question? Thank you.
0
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1answer
817 views

Characteristic function: fourier transform of probability measure or density?

In many texts charecteristic function is defined as a Fourier transform of probability density (if random variable admits a density function). Also we can define a charecteristic function as Fourier ...
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3answers
553 views

are fourier coefficients always symmetric?

this may sound like a dumb question but are fourier coefficients always symmetric? ie $\hat{f}(n) = \hat{f}(-n)$?
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3answers
2k views

Delta function integrated from zero

I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a ...
2
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1answer
289 views

when does derivative of a function coincide with derivative of Fourier series?

Example: for function $$f(x)=x^{3}(1-x)^{3}=\sum f_{s}\exp(2\pi isx)$$ Fourier series of its fourth derivative are different from derivative of its Fourier series $$f^{(4)}(x)=-360x^{2}+360x-72=\sum ...
3
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1answer
1k views

rate of decay of fourier coefficients vs smoothness

suppose $f \in L1$, 2pi periodic and that the fourier coefficients decay with order $|n|^{-k}, k \gt 2$ show that the derivative of f is continuous i read that the rate of decay of fourier ...