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 for triangular wave

Could someone tell me if I've worked this out right? I'm unsure of the process, especially the final parts where I convert it to a sinc function. Please let me know if I've made mistakes anywhere ...
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A property of the Fourier transform

I need a hand proving this property involving Fourier transforms: If we have $F,G\in L^2(\mathbb{R})$, and we denote the Fourier transform as $T$, where $T(F)(\xi)=\int_{\mathbb{R}}F(x)e^{-i\xi ...
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37 views

About Fourier coefficient definitions

I'm studying Fourier analysis and my book gives the following definitions for the Fourier series and Fourier coefficients: Fourier series of $2\pi$-periodic function $f(\theta)$ is defined as: ...
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219 views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
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Uniform Boundedness in N of $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$

Show that $\int_0^\infty \frac{\sin(x)}{x}\,\mathrm{d}x = \frac{\pi}{2}$, and using that show that $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$ is uniformly bounded in N and ...
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In which case should a wavelet transform be applied instead of a Fourier transform?

I was wondering what are the advantages (and possibly the drawbacks) of using a wavelet transform instead of a Fourier transform for the signal processing, are there simple examples to illustrate that ...
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30 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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Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
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31 views

Discrete Time Fourier Tranform of $\frac{sin(n-2)\frac{\pi}{3}}{(n-2)\pi}$

I don't know how to go about solving this. I can see that this can easily be manipulated into the sinc function, but I don't know where that gets me.
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Fourier Transform of inverse powers of the absolute value

I don't think this question has been asked previously, so here goes. I need to evaluate the following integrals - $$ ...
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42 views

Riemann-Lebesgue application

By the Riemann-Lebesgue lemma, I have shown that for any finite interval measurable set $I$ of finite measure, any $h \in \mathbb{R}$, $$\lim_{n \to\infty}\int_I \cos (n(x+h)) \mathop{dx} = 0.$$ I ...
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133 views

Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the ...
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53 views

How to apply Plancherel Theorem here?

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R) and the L^1 norm of f(x) is less than or equal to 8 times (the L^2 norm of f(x)) times the L^2 norm ...
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39 views

use plancherel theorem to prove an integral inequality

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R). I'm sorry I don't know how to use Latex to post the problem. The origional problem is here: ...
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43 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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Reason for restricted frequency range of a DFT

The frequency range of a continuous-time signal extends from $-\infty$ to $\infty$ while for a discrete time signal it's from $-\pi$ to $\pi$ (or 0 to $2\pi$). Why is the frequency range limited for a ...
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Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
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Seasonal adjustment and Fourier analysis

I've been reading up on seasonal adjustment (removing "seasonal" periodic components from a time series) recently and although I see a lot of fancy work around ARIMA models and fancy ways to detect ...
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28 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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Fourier Transform of $sin(5t - \frac{\pi}{4})U(t+8)$

I have this function $$ sin(5t - \frac{\pi}{4})U(t+8) $$ I know the Fourier Transform of $sin(5t - \frac{\pi}{4})$, which is $$ \frac{e^{-\frac{\pi^2}{2}fj}}{2j}\left [\delta (f-\frac{5}{2 \pi}) ...
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Rearrangement of Fourier Series Sum to attain convergence

Let $f$ be a continuous function with diverging partial Fourier sums $S_N(f)(0)$ : $$ f(\theta) = \sum \limits_{k=1}^\infty \alpha_k P_{N_k}(\theta)$$ Let $f(x) \sim \sum \limits_{n=-\infty}^\infty ...
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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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Approximating the Fourier transform with DFT/FFT

Suppose I have a continuous function $f(x)$, $x\in[-L/2,L/2]$. Its $L-$periodic Fourier coefficients are given by $$ \hat{f}[k]=\frac{1}{L}\int_{-L/2}^{L/2}f(x)\exp(-2\pi ikx/L)dx $$ If I apply ...
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Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
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Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
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34 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
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42 views

Fourier analysis how do i calculate an equation

Struggle is an understatement! I'm trying to get my head around Fourier analysis and I have the equation : $$f(x)=2\pi^2+6x^2$$ unfortunately I have no idea where to start and my coursework depends ...
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86 views

How to use Parseval' identity( Plancherel)? [duplicate]

(May be this is very basic question for MO) Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put $$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$ Fix ...
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Fourier transform of a radial function [duplicate]

What is the Fourier transform of 1/|x| in $R^3$? It seems that it is $\sqrt{\pi/2}/|y|^2$ but I am not sure.
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Fourier decomposition of Dirac-delta function under finite limit.

Dirac delta function is said to be Fourier transformation of 1, $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\ e^{i bx} . 1\ =\ \delta(b) $$ This is usually shown by considering the definition ...
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How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
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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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a fourier multiplier inequality

I am reading "Estimates for translation-invariant operators" by Lars Hormander. Currently, I am stuck at the following argument which seems obvious to Lars ))). Let $\varphi(\xi)$ be such a function ...
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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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How does this phase shift in x-space affect the position of a spectrum in k-space?

I'm working on a new form of signal detection with which I hope to recover both the amplitude and phase of a very small signal. However, doing this requires the use of some Fourier maths that I don't ...
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Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
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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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Inverse Fourier transform problem

Can anybody please guide me how to compute the following inverse Fourier Transform ? $$ p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{1}{(1-j\omega\bar{x})^K}e^{-j\omega x}d\omega $$ I shall ...
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48 views

Find coefficients of polynomial that has zeros at certain points

Given a list of values z0, z1, ..., zn-1 (possibly with repetitions), show how to find the coefficients of a polynomial P(x) of degree-bound n + 1 that has zeros only at z0, z1, ..., zn-1 (possibly ...
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How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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For a given sequence $(a_k)$, there is no Riemann integrable function f such that $\hat{f}(k) = a_k \forall k$

I'm working out of Stein's Fourier Analysis: An Introduction, and am on chapter 3. There is an exercise that gives us a specific sequence $(a_k)$ and asks us to show that ...
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Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
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Fourier transform of partial derivative

I am currently doing some reading on using Fourier transforms to solve PDEs and I stumbled upon a property that I am not sure how to prove. Suppose we have a heat-equation $u_t(x,t)=\alpha^2 ...
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Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
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22 views

Spatio-temporal triple correlation

I would like to simplify if possible the spatio-temporal triple correlation of the following function: $$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$ I define the triple ...
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$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
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please help me understand the lecture note? heat equation and fourier series

I don't quite understand equation 3.73 and 3.74. To get $T(x,t)$ I thought I had to multiply F and G. How does that give equation 3.73? I got G as e^{stuff} as in the last bit of equation 3.73. ...
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Heat equation. Find $B_n$ (using boundary conditions?)

Can anyone help me with (b)(i)? I've done the first part of it. I've tried putting some boundary conditions in but cannot find $B_n$ What fact should I use? An infinite slab of material, of ...
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40 views

How to apply the method of stationary phase here?

Consider the following oscillating integral $$ I(\xi;t) = \int\limits_{\mathbb R}\int\limits_{\mathbb R^n} e^{it(\xi y - \theta f(y))}a(y) \, dy \, d\theta, \quad \xi \in \mathbb R^n \setminus 0, ...
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Why is this an inverse fourier cosine transform?

I would like to understand the principle of Fourier transform spectroscopy. This is explained in Wikipedia. I did all the modeling of the system and I got the same formula: $$ I(p,\tilde{\nu}) = ...