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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When is the fourier transform of a quasi-character $\hat c(\alpha)=|\alpha|c^{-1}(\alpha)$?

This is from lemma $2.4.2$ of Tate's thesis. Let $c$ be a quasi-character on $k^{*}$, the multiplicative group of a number field completed at a non-archimedian place. Lemma 2.4.2 For $c$ in the ...
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34 views

on an integral inequality

Let $Q=(-\frac{1}{2},\frac{1}{2}]$ be a unit interval. Prove or disprove the following statement: There is an absolute constant $C$ such that for any Schwarz functions (or smooth functions with ...
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175 views

What's the Fourier Transform of an Error Function?

What is the Fourier transform of $\displaystyle \operatorname{Erf}\left[a+bx^{2}\right] $? I need this in order to evaluate $$ \int_{-\infty}^{\infty}e^{-\beta x^{2}}Erf\left[a+bx^{2}\right]dx $$ ...
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solve equations with exponentials of the unknown

folks, I'm trying to solve $$A = B Y + C Z$$ where $A$, $B$, $C$ are known functions defined on $\mathbb R^3$, the unknowns are basically $z$ defined on $\mathbb R^3$ and $$ Y = ...
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18 views

Limits of the Fourier Integral Transform of a top hat function?

Would the Fourier transform of the following function: be the integration of -(X+1)e^(-ewjt) between the limits of -1 and +1 or -1.5 and -0.5? Would there a shift of limits?
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32 views

Require help with the convolution of two complex conjugates

I need to find the convolution of the following two functions: When rationalizing the denominator, the numerators become complex conjugates of each other. I have tried obtaining the Fourier ...
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How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
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27 views

Discrete Fourier Transform of the infinite series

I am reading this book and having hard time understanding how to get to eq(2) from eq(1) $$P(k,t) = e^{-\alpha t} \sum\limits_{l,m=-\infty}^\infty (-i)^m e^{ik(l+m)} I_l(\alpha t) I_m(i\beta t) ...
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31 views

Fourier series of the function $1-e^{i\delta t}$

Expand the function $1-e^{i\delta t}$ ($t\in [\pi,\pi]$) in a Fourier series relative to the complete orthonormal system $\{1, \cos nt, \sin(n-1/2)t\}$. Fourier series: $$\sum_{n=-\infty}^\infty f_n ...
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53 views

Absolute value of the Fourier Transform of Gaussian random variable

Assume you have a normally distributed random variable $x$ with zero mean $\mu$ and standard deviation $\sigma$. Now you take the Fourier transform of it. The resulting complex random variable ...
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38 views

Rate of convergence of Fourier series

I am having a bit of a confusion regarding convergence results. Suppose $f$ is Lipschitz, or $f \in C^\infty$ and let $S_{N}f$ be its truncated Fourier series. In the wikipedia page ...
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Find $g\in C^\infty(\mathbb R^n)$ one-periodic satisfying the identity below..

Let $f\in L^1(\mathbb R^n)$ and $k\in C^\infty(\mathbb R^n\times \mathbb R^n)$ be a $1$-periodic function in both variables. Is there a $1$-periodic function $g\in C^\infty(\mathbb R^n)$ satisfying: ...
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17 views

decay of coefficients in the expansion into Bessel functions

Let us consider the generalized Fourier expansion into Bessel functions, as illustrated in the Wikipedia page: http://en.wikipedia.org/wiki/Generalized_Fourier_series. Let $J_0 (r)$ be the 0th ...
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31 views

What does an image of Fourier Transformation of an image tell us?

First time studying image processing... I just don't understand what does fourier transformed image of an image describe? For example consider given following pictures, The first one is the image, and ...
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16 views

Relating Fourier Transform to an Integral involving Sin(vt)

I have data for a function $S(Q)$ and I'm trying to find values for a different function $g(r)$ Now I know $g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ$ This is closely related to the sine ...
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17 views

Significance of Mutual Coherence

I am reading about compressive sensing and I am not able to understand the physical significance of mutual coherence. For tow matrices $\Phi$ and $\Psi$, mutual coherence is defined as $\mu(\Phi, ...
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27 views

8 X 8 FFT process

I'm having trouble figuring out how to compute F8 * c using the FFT algorithm. I understand that you factor F8 into ...
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20 views

Inverse Fourier Transform of $S_Y(f)$

I have this power spectral density $$ S_Y(f) =\frac{N_0}{4 \pi ^{2} f^{2}}\left [ 1- \cos(2\pi f T) \right ] $$ Can any one help me how to find the Inverse Fourier transform?
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37 views

slowly varying functions and bounded variation

A function $L:(c,\infty)\rightarrow\mathbb{R}$, with $c>0$ is called slowly varying at infinity in Zygmund's sense, if for x large enough it is positive and for any $\delta>0$ there exists a ...
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17 views

2D Wave propagating in duct with height change

Suppose that we have a two - dimensional rigid wall duct cosisting of two semi - infinite regions $x<0,\ 0\leq y\leq a\ $ and $x>0,\ a<y\leq b$ (this means exactly that there is a height ...
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32 views

Fourier Sine Transform Identity Relation through Integration by Parts

This is purely for my own recreational interest. I've spent the last few days trying to demonstrate to myself that the Fourier Sine Transform and the inverse Fourier Sine Transform return their ...
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54 views

Fourier transform of a function involving $\sec(\omega)$

The summary of my question is: What should I make of $\mathcal{F}^{-1}\left[\frac{\csc(\omega)}{\omega^2-\beta^2}\right]$ where $\mathcal{F}^{-1}$ is the inverse fourier transform (taking $F(\omega)$ ...
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25 views

A question about convolution:

Given $$ P_n(x) = \frac{n}{1+n^2 x^2} $$ and $f(x)$, a continuous periodic function with a period of $2\pi$: Prove that $$ f_n(x) = \frac1\pi\int_{-\infty}^\infty f(x-t)P_n(t)\,\mathrm dt $$ ...
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19 views

Determine whether the set is uniqueness set

We say that $\Lambda$ is a uniqueness set for the Paley-Wiener space $PW_{\pi}$ if $$(F \in PW_{\pi} \wedge F|_{\Lambda}\equiv 0) \rightarrow F\equiv 0.$$ For example, $\Lambda =\mathbb Z$ is a ...
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54 views

calculate the integral using Fourier transform

I am asked to calculate the integral $$\int_{-\infty}^{\infty}\frac{\sin^2(\omega)}{\omega^2}e^{i\omega}d\omega.$$ I read all the posts on this site about the integral ...
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33 views

Inverse Fourier of an Exponential Times Dirac Comb?

What is the inverse Fourier transformation of \begin{eqnarray} \frac{1}{T}\sum_{n=-\infty} ^{\infty} e^{j\pi n} \delta (f-nf_{0}) \end{eqnarray} ? Since $ e^{j\pi n}$ doesn't depend on the ...
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38 views

fourier matlab question

Let's assume that $s(-3)=3$, $s(1)=3$, $s(5)=-1$ $s(-2)=4$, $s(2)=1$, $s(6)=-4$, $s(-1)=4$, $s(3)=2$, $s(0)=-3$, $s(4)=-3$ and $s(k+10)=s(k)$. for this signal the discrete fourier transform ...
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22 views

Frequency Axis problem in a DTFT

I have a doubt related to calculating the Discrete Time Fourier Transform (DTFT) by hand. Specifically in how calculate the frequency axis of the spectrum. My signal has N values and was sampled at FS ...
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30 views

Thinking about a convolution on the circle as a convolution on the real line

I hope my question is not too vague. Let $f$ and $g$ be (smooth, say) real-valued functions on the circle. Let $f*g$ denote the standard $L^2$ convolution on the circle. Question: is it possible to ...
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The integral equation satisfied by Fourier transform of an ODE

In my mathematical methods exam I came across the following question: Show that the Fourier transform of the following differential equation, $$ x\frac{dy}{dx}+e^{-x^2}y=0 $$ is as follows $$ y(k) ...
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37 views

Convolution + fourier

So let's say we have $s(t)=e^{-a|t|}$ so I calculated that $\hat{s}=\frac{2a}{a^2 + 4 \pi^2 v^2}$. Then if $h_a = \frac{2a}{a^2 + 4\pi^2 t^2}$. using $\hat{s}$, how can we prove that $h_a * h_b = ...
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Does there exists $f\in L^{1}(\mathbb R)\cap FL^{1}(\mathbb R)$(=Fourier algebra) but $|f|\not \in A(\mathbb R)$?

For $f\in L^{1}(\mathbb R)$; We define the Fourier transform of $f$ as follows: $$\hat{f}(\xi)= \int_{\mathbb R}f(x) e^{-2\pi i \xi x}dx; \ \text {for} \ \xi \in \mathbb R.$$ Consider a Fourier ...
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Bounded Operator, Proof

Let $V$ be continuous, $V\geq0$ and $V\rightarrow \infty$ as $||x||\rightarrow \infty$. Define $H:=-\Delta+V$. I want to show that $$G:=\big((-\Delta)^{\frac{1}{2}}+1\big)(H+1)^{-\frac{1}{2}}$$ is ...
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62 views

A estimate for the L1 norm of the Dirichlet kernel

Consider the classical Dirichlet kernel given by $D_{N}(x) = \displaystyle\frac{sin((2N + 1)\pi x)}{sin(\pi x)}$ for $x\neq [-\frac{1}{2} , \frac{1}{2}]-\{0\}$ and $D_N(0) = 2N +1$ . Define $L_N = ...
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Is the Fourier series also a Laurent series?

For a holomorphic function $f(z) = \sum^{\infty}_{n=-\infty}{a_nz^n}$, the substitution $z = e^{j\pi\omega}$ yields the Fourier series $f(z) = \sum^{\infty}_{n=-\infty}{a_ne^{j\pi\omega n}}$. Would it ...
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35 views

simple question Fourier series :even fourier coeficients of a even function

Let $f \in L^{1}(R)$ a even function of period 1. The Fourier series is defined by $$ \displaystyle\sum_{n=1}^{\infty}\hat{f}(n) e^{2 \pi n i x}$$ where $\hat{f}(n) =\displaystyle\int_{0}^{1} f(x) ...
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Boundary of real part of functions in $H^p$ and Poisson nontangential maximal function

I have two questions when reading on $H^p$ spaces, many books do not give their proofs. First we reminde that $H^p(\mathbb R^2_+)$ consists of all functions $F$ which is analytic in the upper half ...
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17 views

Convolution of an image with a kernel that is a product of two functions

Suppose that $G(i,j)$ is a Gaussian decay function on the distance between points $i$ and $j$ of an image. In addition, $D(i,j)$ is the difference between the VALUES of the image at those points. ...
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35 views

DFT Zero-padding : what prepending with zeros does?

I am studying Fourier transform and want to understand better some point regarding zero-padding in DFT. All known sources say that padding is done by appending the data with zeros. However, if I add ...
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DFT detemine frequencies

I am just begining learning about DFT and I am bit unsure on what is happening mathematically during DFT. I've sampled a signal , a sine wave $sin(1000\cdot 2\pi t)$. And performed DFT to calculate ...
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Iteration of Fourier coefficient.

I would like to know if there's any work or if it's even interesting to look at an iteration of fourier coefficients. Here's what I mean, take $f\in L_2(-\pi,\pi)$ for simplicity, and compute its ...
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Derivative of squared Fourier transform

I haven't found any relative to this, so I would like to get some help. I have a function $h(x) = |\mathcal{F} [P(x) e^{ic+iZ(x)a}]|^2 $ and I would like to find the derivative with respect to the ...
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17 views

Polar form of Bi-Quatenrion

I have a complex Quaternion(Bi-quaternion) and i want to convert that to Polar form (Euler). Let say we have Fourier transform of a ( bi-quaternion ) like, then how can we get a polar form ...
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134 views

Difficult Fourier transform

While looking at non-local modifications to wave propagation in 2d I have run into the following integral $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega dk \ln(k^2-\omega^2)e^{-i\omega ...
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28 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
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35 views

DFT one-dimensional vector

That's a way to define one-dimensional Discret Fourier Transformation (DFT)? If I have a signal $x \in \mathbb{R}^N$ and I take a rectangular window of lenght M: $y = (x_m, x_{m+1}, x_{m+2}, .., ...
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estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
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110 views

Trying to figure out Fourier transform of {(0.5^n)(u(n))

I'm working in a signals class for continuous signals, and we have this problem shown above. I have tried using this function f_1 X f_2 = F_1 * F_2, where I'm ...
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51 views

Possible values of characteristic functions (Fourier transforms)

Can a characteristic function $\varphi_{X}(u)$ from probability theory (the Fourier transform of a probability measure) ever equal zero for either any value of $x$ or any value of $u$? This has been ...
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How do I solve for the down sampled image after a discrete fourier transform?

Let $x_z[n], n \in \{0, . . . , 2N − 1\}$ be a one-dimensional image of length $2N$ with zeros at every alternate pixel. That is, $x_z[n] = 0$ for every odd n. Now suppose we down-sample $x_z[n]$ by a ...