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 takes point wise multiplication to convolution multiplication

Let $f, g\in S(\mathbb R^{n}) (L^ {1}{(\mathbb R^{n})} ) $. It is well known that, $ F(f\ast g)= \hat {(f\ast g)} = \hat f \cdot \hat g = F(f) \cdot F(g)$, (that is, Fourier transform takes ...
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137 views

Integration by parts argument for the Fourier Transform of a Schwartz Function

Here is a part that I am having hard time to understand. Actually, it is basically the general trick in Schwartz space that is just integration by parts. However, I would like to see how we get rid ...
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91 views

Fourier transform of $(x^2+b^2)^{-1}$

The question asks to find the Fourier transform of $(x^2 + b^2)^{-1}$ given that $F[e^{-b\vert x\vert}](k) = \frac{1}{\sqrt{2\pi}}\frac{2b}{k^2+b^2} $ $$\sqrt{\frac{\pi}{2b^2}}e^{-b\vert x\vert} = ...
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96 views

Inverse Fourier-cosine transfrom

Suppose we have a function $F(x)$ given by the integral: $$F(x)=\int_{0}^{\infty}f(t)\frac{\cos(t\log x)}{t}dt\;\;\;\;\;(x>1)$$ This looks tantalizingly like a Fourier-cosine transform of ...
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30 views

Laplacian for Radon inversion theorem

Can someone check my proof in regards to the inversion of the Radon transform in $\mathbb{R}^2$ and $\mathbb{R}^n$. define $(-\Delta)^a f(x) = \int_{\mathbb{R}^d} (2\pi|\xi|)^{2a} \hat{f}(\xi)e^{2\pi ...
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40 views

Is there an expansion for element-wise scaled convolution?

If $x = a\cdot b$ is used to indicate $x_i = a_i\cdot b_i$, $y = a / b$ denotes $y_i = a_i / b_i$, and $a*b$ denotes convolution, then is there a simplification for this expression: $$ ...
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49 views

The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions).

Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries \begin{equation} r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; ...
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103 views

Bluestein Algorithm for Fast Fourier Transform

Can anyone demonstrate the full algorithm of Fast Fourier Transform? Because from Wikipeida and other internet sources, I saw that there are different ways of padding. So can anyone tell me when the ...
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32 views

Fourier transform of $\frac{1}{r^k}D^t U$ in terms of the Fourier transform of $U$

What is the Fourier transform of $\frac{1}{x^2}D^4 U(x)$ in terms of Fourier transform of $U(x)$. Here $D$ is the derivative of $U$ with respect to $x$, and $U(x)$ is the velocity function dependent ...
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31 views

Best transform for detecting frequencies of sinusoidal signals.

Between Discrete Fourier Transform (DFT) and Discrete Cosine Tranform (DCT) could we say that one of them is better than the other in detecting frequencies of sinusoidal signals?
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23 views

On linear translation commutative operators

I have a problem on linear translation commutative operators. Assume, that $T$ is linear on complex-valued real functions and $T\tau_\alpha = \tau_\alpha T$, where $\tau_\alpha x(t) = x(t-\alpha)$. In ...
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52 views

approximate Fourier transform

Let $\mathcal{F}$ stand for the Fourier transform. Suppose $f : [-\delta/2,\delta/2] \to \mathbb{C}$ is a "nice" function. Is it true that $$\left|\mathcal{F} \left(e^{imx} \left(e^{ix^2}-1 ...
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68 views

2D Spatial - Frequency Domain Results Not matching?

I've got a process I'm performing in the spatial domain as: 1. Create a grid of points, 2. Take a square of some Length and Width and place it on the x-y axis origin 3. Count the number of points ...
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90 views

Fourier, the Fourier transform

Could You help me? Where $g(t)$ is Cantor function: $$G(\omega)= \int_0^1 e^{2\pi i\omega t}dg(t)$$ Show, that $G(\omega)\not\to0$, if $\omega\to\infty$
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68 views

Inverse fourier transform of product of exponential and hyperbolic sine/cosine

I would like to get the IFT of (with respect to $k$): $$\exp\left(-\frac{t}{2}(\lambda+k^2D)\right)\cdot \left[ ...
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40 views

FFT can be computed in $\frac{N}{2}\log N$ complex multiplies and $N\log N$ complex additions?

According to Wikipedia's page on FFT, FFT can be computed in $\frac{N}{2}\log N$ complex multiplies and $N\log N$ complex additions. Why not in $N\log N$ complex multiplies and $N\log N$ complex ...
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32 views

Fourier transform of $f(r,r',\theta,\theta')$

How can I calculate the FT of: $$\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,f_n(r,r')=\sum_{n=-\infty}^{\infty}\,e^{in(\theta' -\theta)}\,\frac{J_n(\alpha r)J_n(\alpha r')}{[(\alpha ...
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43 views

Intuitive way to understand the triangle spectrum?

Image on the top is in the time domain, image on the bottom is in the frequency domain. What do we see $-2T$ and $2T$ on image in the time domain and why do we see $-\frac{1}{2T}$ and $\frac{1}{2T}$ ...
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38 views

Square equivalent of $circ(r)$

I would like to know if there is a similar function to $$circ(\sqrt{x^2+y^2})=1 , 0\leq \sqrt{x^2+y^2}\leq 1$$ but with a square domain $0\leq x\leq 1$ and $0\leq y\leq 1$. If yes, which is its ...
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27 views

Show that$ a$ is a differential of order $m$.

Lat $a = a(x,\zeta) \in S_{1,0}^m(\mathbb{R}^n,\mathbb{R}^n)$. Write $n=n_1+n_2$ with $n_2\geq 1$ and $\zeta = (\zeta_1,\zeta_2)$ with $\zeta_i\in \mathbb{R}^{n_i}$. Suppose that $a$ does not depend ...
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28 views

Signed amplitude spectrum

I have never seen a "signed amplitude spectrum" of a Fourier transform in the literature. Let $X(\omega)$ be the Fourier transform, which can be represented as the product ...
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63 views

Intregral of exponential of Shannon Entropy Function

Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of $F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$ ...
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144 views

Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.

Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$. [Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
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46 views

Fast Fourier Transform of 2D axisymmetric geometry

I need to compute the FFT of a signal in 2D axisymmetric geometry. The signal consists of a snapshot in time of a laser beam for which I have values in z (direction of propagation) and r (from 0 to ...
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48 views

Question regarding Fourier transform

Show that if $ f \in L^1[(0,2\pi)]$ and $\sum^\infty_{-\infty} |\hat{f}(n)|^2 < \infty$ then $f \in L^2[(0,2\pi)]$ where $\hat{f}(n)$ is the Fourier transform I started by by substituting the ...
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41 views

DFT of some functions

Given the DFT pair $x[k]$ and $X[r]$, for a sequence of length N, express the DFT of the following sequences as a function of X[r]: $$ y[k]=x[2k]$$ I guess this is a simple question, but I can't ...
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166 views

Fourier transform of $\log(f(x))$

Suppose for a function $f(x)$ becomes $F(k)$ after a Fourier transform, what is the Fourier transform of $\log(f(x))$? I cannot find any related formula in Fourier transform table or list properties. ...
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46 views

Fourier Transform in 3-dimension

I am trying to solve this integral in 3-dimensions $$ \int_{\bf{k}} \frac{({\bf q}/2-{\bf k})e^{i{\bf k}\cdot{\bf r}}e^{(-k/\Lambda)^4}}{({\bf q}/2-{\bf k})^2+m^2}.$$ Any suggestion??? Thanks
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137 views

Heat Equation - with boundary conditions?

I have a question concerning the heat equation: Formulate the homogeneous Cauchy problem for the heat equation on $\mathbb{R^n}$, and give uniqueness and existence results, including a solution ...
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38 views

Is it even possible to “visualize” this in the time domain?

I'm trying to understand single sideband modulation. If you want to conserve bandwidth and you don't mind complicated math, then SSB is for you. So far the only thing I could find online to help me ...
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157 views

Autocorrelation derivation using fourier transform

I am stuck with basic understanding of the Auto-correlation derivation of a simple signal and I would be pleased if you could help me out with that. Lets have a signal $x(t)=\cos(2\pi{f_{0}}{t})$. ...
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47 views

Is $C(B^c)$ an open set?

Assume that $B$ is an open set, if $C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,\sum_{i=1}^{n}\lambda_{i}=1,x_i \in B\}$ is a convex that contains $B$, $C$ is an open set? What's ...
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How to show that if $g_{\epsilon}(x) = \sum_{m=1}^M = f(m/M)\chi_{[(m-1)/M,m/M)}(x)$ and $x \in [(m-1)/M,m/M)$ then $g_{\epsilon}(x) = f(m/M)$?

How to show that if $g_{\epsilon}(x) = \sum_{m=1}^M = f(m/M)\chi_{[(m-1)/M,m/M)}(x)$ and $x \in [(m-1)/M,m/M)$ then $g_{\epsilon}(x) = f(m/M)$ and not $g_{\epsilon}(x) = \sum_{m=1}^M f(m/M)$? ...
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68 views

sum to integral inequality step in a proof of Kolmogorov

If I have $N$ numbers $x_j$ very very close to $N$th roots of unity. How could I show $$\frac{1}{N} \sum_{j=1}^N \left|\sin(\tfrac{1}{2}(t-x_j))\right|^{-1} > \int_{1/N}^\pi ...
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52 views

Fourier Transform vs. Sum of single sines

I have a table of frequencies ($f$) and corresponding Amplitude ($A$) as well as phase ($p$) values. Unfortunately the result of performing an iFFT on these data, $$s_1(t) = ...
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104 views

Wavelet Transforms

I don't know as much as I would like to about Fourier analysis and I know almost nothing about wavelets. So just have a few conceptual questions to determine whether I should pursue their study or ...
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114 views

Fourier Series and Filter in Function

Let $L=1, A=1$ and $f\in L^2([-L/2, L/2])$, with Fourier series $$f^{t}=\sum_{n=-K}^{K}a_n \exp(2j\pi xn/L),$$ truncated at $K$. Has this function, $f^{t}$, any relation with Fourier Inverse ...
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115 views

Taking inverse Fourier transform of complicated multipart equation

Define $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} ...
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312 views

Fourier Series Coefficients for Signals

The question is: We specify the fourier series coefficients of a continuous-time signal that is periodic with period 4. Determine the signal x(t). $a_k=\begin{cases} 0, & k=0\\ ...
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98 views

Discrete fourier transform and Frobenius norm

I have thousands of data files which are essential DFT data of plots such as this: And a DFT of the plot with a "Hard" threshold of 0.9 gives me: This DFT is just the left top corner of the DFT ...
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83 views

Fourier series of function defined by trigonometric series.

I'm dealing with the problem, that the function defined by trigonometric series(that is, limit of symmetric sum of $e^{inx}$) I have shown that this function converges everywhere in pointwise sense. ...
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76 views

Filter/Removal of periodic 'delta-peaks'

Currently I am measuring data (Counts over Time). Due to measurement problems I have some nasty peaks in this data. These peaks are periodical, very sharp (~3 datapoints over a range of 10000) and ...
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40 views

Is there meaning behind a sum of Derichlet kernels?

I have arrived at the following expression: $$\sum_{n=0}^{\infty} \sqrt{n+1}D_{n}\left(x\right)$$ Where $D_n\left(x\right) = ...
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231 views

fourier transform of positive function

I am having trouble with this question: Show that there exists a compactly supported $C^\infty$ function $\phi$ on $\mathbf{R}$ such that $\phi \ge 0, \phi(0) >0$, and $\hat{\phi} \ge 0$. I ...
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finding the fourier coefficients of $f(x) = \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}$

This is what i know so far: The given series uniformly converges by the M-test and that i can swap the integration and the sum when calculating the coefficients. Apparently i am supposed to use the ...
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300 views

Fourier series of $f(x)=e^{\cos(x^2)}$

I would like to write the $2\pi$-periodic function $f(x)=e^{\cos(x^2)}\; $ $(0 \leq x \leq 2\pi)\;$ as a Fourier series, but I am unable to carry out the integration. In order to write it as a Fourier ...
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406 views

Convolution between a kernel and an image with FFT

In the FFT2D paper (Fast Fourier transform used for a convolution with a kernel in the frequency domain), I'm lost at the second page first picture: ...
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89 views

How to illustrate the transfer function with a given equation?

Homework/Revision question: Define the transfer function of a linear system. Illustrate your answer by considering the system governed by the equation: $\frac{dy}{dt}+ay=bx$ where x and y are ...
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96 views

What happens to Fourier Transform of function when the function's time scale is changed?

When a function $f(t)=exp(-|t|)$ for example undergoes Fourier Transformation, it gives $F(w)=\frac{-2}{1+w^2}$ But what happens to the result if the time scale is scaled and shifted, so that $t ...
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360 views

How to use Fubini's theorem when integrating over Euclidean Space

I am currently studying the Fourier transform in Euclidean Space $\mathbb{R}^n$, using Knapp's book "Basic Real Analysis". Upon proving the property that the Fourier transform turns Convolution in ...