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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reducing amplitude of fft spectrum with constant phase

I have a time series that I converted to the frequency domain using fft in matlab. I want to reduce the amplitude at a given frequency range (remove a peak between 1.5 and 2Hz) but keep the phase ...
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193 views

Fourier expansion of sine of cosine function

What is the Fourier expansion of $$\sin\left(A\cos(\omega t)\right),\qquad 0<A<1,$$ in the frequency $\omega$ domain?
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469 views

Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
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1answer
915 views

Decay rate of Fourier coefficients of a continuous function with discontinuous derivative

Can you prove or disprove the following: The Fourier coefficients of a continuous function with discontinuous derivative decay like $ 1/n^2$.
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0answers
498 views

Shannon vs dirichlet kernel interpolation method for signal reconstruction

I am currently studying fourier transform, and especially the way that the signal could be reconstructed from its spectrum. In many lectures, I have seen the shannon interpolation method to ...
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1answer
448 views

Show that the Fourier transform of a radial function $ L^1 (\mathbb{R}) $ is also radial

How do I prove that the Fourier transform of a radial function $ f \in L^1 (\mathbb{R}) $ is also radial function? I tried by polar coordinates but I dont got.
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745 views

How to justify the solution of this problem?

Assume $\mathbf{x} \in \mathbb R_+^N$ with support $P=\{p_1,p_2,\cdots,p_K\}$ ($P$ is unknown). We already know that $$f_1(\mathbf{x}) = f_2(\mathbf{x}) = \cdots = f_{N-1}(\mathbf{x})$$ where ...
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2answers
247 views

Conceptual question about Discrete Fourier Transform

On the wikipedia page for the discrete Fourier transform, the first sentence says: In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a ...
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120 views

inverse Radon transform

I'm having a trouble understanding a proof in regards to the inversion of the Radon transform in $\mathbb{R}^3$. The statement is as follows: if $f \in \mathcal{S}(\mathbb{R^3})$, then ...
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2k views

Proof of Fourier Inverse formula for $L^1$ case

I know this may be a stupid question, but still hope someone can help me. I was trying to prove the Fourier inversion formula for which $f$ and $\hat{f}=\int_{\mathbb{R}}f(x)e^{-i2\pi xy}dx$ both lie ...
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1answer
92 views

Conditions for the Fourier transform of an $L^1$ function to be in $L^1$

I know that if one needs to use the Fourier inversion formula, one condition is to make sure the function $f$ and its Fourier transform $\hat{f}$ both lie in $L^1$. If now I have a function $f$ which ...
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3answers
394 views

Further studies on Fourier Series and Integrals.

If you had to choose two books from the following list, which pair would you chose, and why? If you haven't read any, would you pick any pair among the list based on the author of the book? I am ...
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41 views

Fourier transform of a sequence by a matrix

Let $n$ be a positive integer and $H$ the Hilbert space $\ell^2(\mathbb Z^n,\mathbb C^n)$. For $u\in H$, denote by $\mathcal{F}(u)$ the Fourier transform of $u$, defined by $\displaystyle ...
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1answer
99 views

Pade approximatn and Fourier integrals

we know that the Fourier integrals $$ \int_{-\infty}^{\infty}dx x^{m}e^{ixu} =F(u)$$ exists only in the sense of distribution however what would happen if we approximate the complex exponential by ...
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2answers
337 views

isoperimetric inequality using Fourier analysis

I'm trying to prove an isoperimetric inequality, but I have absolutely no idea how to go about it. let $\Gamma$ be a closed plane curve parametrized by $\gamma(t) = (x(t), y(t))$ on $[-\pi, \pi]$. ...
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0answers
32 views

An alternative proof without multiplier theorem

Given $f\in \mathcal{S}'$ and $\phi\in C_0^\infty$, with $\operatorname{Supp}\phi = \{1 \leq \xi \leq 2\}$. Define $\phi_j = \phi(2^{-j}\cdot)$. We have if $\phi_k*f\in L^p$, then for all ...
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1answer
379 views

Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$

The sobolev-space $H^s([-\pi,\pi])$ can be embedded into $C^{(\alpha)}([-\pi,\pi])$ (space of $\alpha$-Hölder-continuous functions) and vice-versa. My question is for which exponents $s, \alpha$ can ...
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2answers
304 views

Fourier transform of $e^{-|x|}$

I am looking for the Fourier transform on $\mathbb{R}^3$ of $e^{-|x|}$. I tried in spherical coordinates with $x=(r,\phi,\theta)$ and $\xi=(|\xi|,\phi_2,\theta_2)$: $$\int_{\mathbb{R}^3} e^{-|x|} ...
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88 views

The Envelope (wave front) of a span of plane waves.

Imagine we have for every real direction $\mathbf n \in \mathbb R^3$ a plane wave, be it either quantum, eletric, magnetic, acoustic or in my case some sheets of paper I am holding up. Each of these ...
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2answers
86 views

Compute $\int_{\mathbb {R}}\sin \left(\frac {\pi x}{x^2+1}\right)\frac{1}{x^2+1} dx$

How do I compute this integral? $$\int_{\mathbb {R}}\sin \left(\frac {\pi x}{x^2+1}\right)\frac{1}{x^2+1} dx$$
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72 views

elliptic operator and wave front set

Let us $f(x) \in C^\infty $ on $\mathbb{R}^n$, and the pseudo-diff. operator $ Q$ is defined by: $(Qu)(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}e^{ix\xi }f(x)\left | \xi \right |\hat{u}(\xi) d\xi$ Where ...
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78 views

associated haar measure of dual group of a product

Let $G_1$ and $G_2$ be locally compact abelian groups equipped with Haar measure $\mu_{G_1}$, $\mu_{G_2}$ and $G=G_1 \times G_2$ equipped with $\mu_{G_1} \otimes \mu_{G_2}$. I would like to show that ...
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1answer
46 views

$ \int_{-\infty}^{\infty} \frac{|\widehat{\psi}(\omega)|^2}{|\omega|}d\omega = C_{\psi} < \infty \implies \widehat{\psi}(0) = 0$

Let $\psi \in L^2( \mathbb{R})$ and suppose that it satisfies the admissibility condition $$ \int_{-\infty}^{\infty} \frac{|\widehat{\psi}(\omega)|^2}{|\omega|}d\omega = C_{\psi} < \infty $$ where ...
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1answer
851 views

How to solve differential equations using fft?

Can anyone point me to the principles and books/websites about it? Which properties must the differential equation have that a solution with fft is possible? Why can it be solved that way?
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118 views

Clarification on “Inverse Fourier transform of $(1 \pm \hat{f}(w))^{-1}$”

The original question: Assume I have a real function $f(t)$ with Fourier transform of $\hat{f}(w)$. Can one say anything about the inverse Fourier transform of $\frac{1}{1\pm\hat{f}(w)}$? ...
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1answer
95 views

Can one zero-pad data prior to Fourier transformation, then reverse the change afterwards?

Suppose I have a set of $n$ points $\underline{x}\in\mathbb{C}^n$ with $n \in \mathbb{P}$ ($n$ is prime), and I want to find the Fourier transform of $\underline{x}$. There are some prime-length ...
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1answer
47 views

Determine negligible coefficients in spectrum

Suppose I have some function $f$ that I have sampled at $N$ points and I preform a transform on it (this could be a Fourier transform, or perhaps a Hadamard, or really anything eles - I'm hoping for a ...
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1answer
341 views

Unsure How to Take This Fourier Transform's Inverse?

I have to convolve the following signal and its impulse response, and I thought taking the Fourier Transform would be the best approach: $$x(t) = te^{-2t}*u(t)$$ $$ h(t) = e^{-4t}*u(t)$$ Where ...
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364 views

When does Discrete Fourier analysis fail to detect a frequency?

I'm using python to learn about Discrete Fourier Analysis. What I want to understand is when does the technique fail to recover some frequency of the signal? I understand how this can occur via the ...
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2answers
747 views

Fourier transform of a function of absolute values

I need help in computing the Fourier transform of $$\frac{1}{1+|x|^2}$$ with $x$ scalar. I tried using Cauchy integral formula with contours centred around the origin but I get stacked.
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134 views

Deconvolution of a convolution product with $Ax\ /\ (x^2+l^2)^{3/2}$

This is not a homework, and I have no idea whether it could be one. It is only a request for help, as I do not have any experience using Fourier transform. The origin of the problem is from physics. ...
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330 views

Inverse Fourier transform of $(1\pm\hat{f}(w))^{-1}$

Assume I have a real function $f(t)$ with Fourier transform of $\hat{f}(w)$. Can one say anything about the inverse Fourier transform of $\frac{1}{1\pm\hat{f}(w)}$?
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90 views

Equidistribution of sequence $an^b$, 0<b<1, a>0

This is Exercise number 8, Chapter 4 in Stein's book on Fourier Analysis. I'm supposed to solve it by giving an asymptotic bound to the exponential sum created from Weyl's Criterion, but so far have ...
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1answer
3k views

How is the Inverse Fourier Transform derived from the Fourier Transform?

Where does the $\frac{1}{2 \pi}$ come from in this pair? Please try to explain the Plancherel's theorem and the Parseval's theorem! $ X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$ $ ...
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775 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
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1answer
195 views

Fourier Transform of short pulse

I'm trying to take the fourier transform of a short laser pulse, represented by $E(t) = E_oe^{-(t/\Delta T)^2}\times e^{-i \omega t}$ E is the electric field of the laser pulse. $E_o$ and $\Delta T$ ...
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3answers
297 views

Intuitive explanation of a flat DFT

Given a vector $${\bf x}= [a, 0, \dots, 0], a \neq 0, {\bf x} \in \mathbb{R}^n$$ If I compute its Discrete Fourier Transform (DFT), I get $${\bf DFT}( {\bf x}) = [a,a, \dots, a]$$ I.e., a vector ...
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63 views

What sequence has this Discrete Fourier Transform?

Suppose $$ x[n]= \begin{cases} x_i &, i \in P\\ 0 &, i \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_i \geq 0$. Suppose these equalities hold : $$ ...
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115 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
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3answers
308 views

Recovering the spatial Fourier transform from the space-time Fourier transform

This CW question is aimed at developing some intuition (grokking) about a certain formula of Fourier analysis. Any kind of explanation (physical, geometrical, analytical ...) is welcome. If we ...
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2answers
912 views

Programming discrete fourier coefficients in matlab

Alright so I am having the following issue: I want to figure out how to find the fourier coefficients of the following function: $$D(X)=\frac {a'(x)} {1+a'(x)^2}$$ Where $a(x)$ is an arbitrary ...
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123 views

The algebra of Fourier transforms (and summations)

In a given system, the equation for potential energy is $$P_{x_j}=\frac{(x_{j+1}-x_{j})^2 + (x_{j}-x_{j-1})^2}{2}$$ and the equation for kinetic energy is $$Q_{x_j}=\frac 12 \dot{x}_j^2.$$ These ...
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72 views

Fourier Coefficient Manipulation

Suppose I have a function $f(x)$ which has fourier coefficients $$f_m=\frac 1 d \int^d_0f(x)e^{im\frac {2\pi} dx}dx$$ Which in discrete form (fft) is: $$f_m=\frac 1 N\sum ^Nf_ie^{im2\pi x}$$ I then ...
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Laplace equation Fourier transform

I tried to solve but I didn't. Please help me. $$u_{xx} + u_{yy} = 0$$ $$-\infty < x < \infty$$ $y>0$ , $u_{y}(x,0)=f(x)$. Show that $$u(x,y)= \frac{1}{2 \pi} ...
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214 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
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1answer
468 views

How to extract specific frequencies in Discrete Fourier Transform.

I'm collecting accelerometer data and interested in extracting frequencies from 1-10 Hz. I'm aware of how to do the FFT but not sure how to extract these frequencies 1Hz, 3Hz and 10Hz. Any pointers?
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344 views

Solving Poisson's equation for $\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) \, \delta(y)$

Problem statement I took an exam, where I had the following task: Determine the electrostatic potential for the charge distribution $$\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) ...
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50 views

A Fourier Coefficients (Series) problem

Let $\alpha$ be a number such that $\alpha/\pi$ is not rational. Prove that (1) $$\lim_{N\to\infty}\sum_{n=1}^{N} e^{ik(x+n\alpha)}=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{ikt}\,dt,$$ (2) for any ...
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1answer
134 views

Parseval's Theorem on a Random Signal

I'm struggling with Parseval's Theorem. I'm trying to relate variation in the time domain to the average value in the frequency domain. To do this, I'm performing the Fourier Transform on an arbitary ...
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1answer
100 views

Are the Fourier sine and cosine transforms injective?

Are the Fourier sine and cosine transforms defined by $$\mathcal{F}_s[f(x)](t)=\int_0^\infty f(x)\sin(x t)\text{d}x$$ and $$\mathcal{F}_c[f(x)](t)=\int_0^\infty f(x)\cos(x t)\text{d}x$$ injective? ...