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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1answer
399 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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0answers
46 views

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 ...
2
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1answer
129 views

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 $\sum\nolimits_{k=-\infty}^{\...
3
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0answers
159 views

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 ...
2
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0answers
146 views

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 u_{xx}(x,...
2
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2answers
43 views

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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0answers
100 views

Spatio-temporal triple correlation

I would like to simplify if possible the spatio-temporal triple correlation of the following function: $$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$ where $\delta$ is the Dirac ...
3
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0answers
65 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

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, |f|f,...
0
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1answer
54 views

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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2answers
47 views

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 ...
2
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1answer
85 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, \...
1
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0answers
40 views

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}) = I(\...
0
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2answers
72 views

Sampling and reconstruction frequency

For a sinusoid frequency of $1200$Hz and a sampling frequency of $2000$Hz and a reconstruction frequency of $2000$Hz and $3000$Hz, what frequency will the sinusoid be after the reconstruction? ...
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0answers
82 views

Inverse Fourier transform of two variable function $F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}$

I am trying to find the inverse Fourier transform of: $$ F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}, $$ where $k^2 = k_x^2 +k_y^2 +k_z^2 = k_\rho ^2 +k_z^2 $ is a constant. I am getting confused as to ...
0
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1answer
66 views

Fourier transform real and imaginary part question?

I have to find the fourier transform of $f(t)=e^{-a^*t}*u(t)$ For a>0 the signal has an infinite value therefore doesnt have a Fourier transform.For a>0 we have: $F(w)=\int_0^∞[e^{-a*t}*e^{-j*w*t}]\,...
0
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1answer
48 views

Inner product of functions is preserved by inner product of fourier co-efficients, more to plancherel theorem

By Plancherel theorem the map $T:L^{2}(\mathbf{T}) \longrightarrow l^2(\mathbf{Z})$ defined by $T(f) =(f^{\wedge}(n))_{n \in \mathbf{z}}$ is a surjective isometry. But I have to show a bit more.That ...
0
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1answer
68 views

Fourier series/transform with cutoff

(1) The span of the trigonometric polynomials is $span \{ e^{i n x}\ |\ n \in \mathbb{Z} \} = L^2([-\pi,\pi])$. Is there any nice way of characterizing the span with a cutoff, namely: $span\{ e^{i n ...
0
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2answers
74 views

Very short theory question signals?

My teacher asked us this question yesterday in the lecture but it didn't make any sense to me. He asked: What do the coefficients of the exponential Fourier series represent? Also, what's the ...
6
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1answer
71 views

How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
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0answers
84 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
4
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1answer
186 views

What happens if I remove the sign from the exponential of Fourier Transform?

Forward Fourier Transform $\hat f(x)$ is defined such as $$ \hat f(x) = \int_{-\infty}^{\infty}{f(t) e^{-2\pi t x i} dt} $$ but I am wondering what happens if I define it $$ \hat f(x) = \int_{-\...
4
votes
3answers
77 views

Why does $\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\|^2 = \sum_{i=1}^{N}|\langle f,\phi_i\rangle|^2 $?

I'm reading about Fourier analysis and there is one equality, which I don't understand. Why does: $$\left\|\sum_{i=1}^{N}\langle f,\phi_i\rangle\phi_i\right\|^2 = \sum_{i=1}^{N}|\langle f,\phi_i\...
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0answers
52 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral $$...
0
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1answer
152 views

2-D Fourier Transform of complex exponential with 2-D quadratic phase

I've been looking around to see if there is either an exact transform pair or an approximation to either of the following but have not been able to find anything: $$ \mathcal{F}_{xy}\left( e^{i\cdot(...
2
votes
1answer
113 views

Big O proof of Fourier Coefficient

Let $f(x)$ be a $2\pi$ periodic function on R. Assume that Hölder continuous: $$\sup_{x\neq y} \frac{|f(x)-f(y)|}{|x-y|^{-\alpha}} \leq C$$ for some constants $C$ and $\alpha \in \,]0,1]$. Prove ...
3
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1answer
131 views

Why should we use the Fourier Transform?

I'm a CS/Math double major, and during my study (and reading sources out of my own interest) I've had some encounters with the Fourier Transform. I understand the theory behind Fourier series, and ...
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0answers
47 views

Fourier transform evaluation

Let $x\in \mathbb{R}^n$ and $h(x)=\left\{\begin{array}{cc} 1 & ,|x|<1\\ 0 & ,|x|\geq 1 \end{array}\right.$ I want to find the Fourier transform , $\hat{h}(\xi).$ Here is how I proceed $...
3
votes
2answers
132 views

Why are almost all prime sized circulant matrices non-singular?

Consider an $n$ by $n$ circulant matrix whose values are either $1$ or $0$. Now let $n$ be a prime. For any such $n$ there are exactly two circulant matrices that are singular (over $\mathbb{R}$). The ...
3
votes
1answer
138 views

Why a periodic function can be expressed by a set of finite numbers — question about Fourier transform?

For a periodic function $f(x)$ with period $p$, Fourier transform says that it can be "expressed" by a set of infinite many number. $$ f(x)=\sum_{k=-\infty}^\infty F[k]e^{2\pi i kx/p} $$ where $$ F[...
1
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1answer
90 views

Fourier Series of a decaying Cosine

I'm trying to find a handy way to find the infinite sum $ \sum_{n=1}^\infty \frac{cos( a n)}{n^2} $ through a Fourier series. The regular sum can be evaluated using Mathematica fairly easily, and ...
0
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1answer
112 views

Solving the heat equation using Fourier series; specific questions

Like this previous question, Solving the heat equation using Fourier series, I too am reading the same wikipedia article, http://en.wikipedia.org/wiki/Heat_equation#...
1
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1answer
146 views

Calculate step response from impulse response LTI-system.

Can someone please give me a few pointers on how to calculate the step response for an LTI system with this impulse response?? \begin{equation} h[n] = 2^nu[n]. \end{equation}
3
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0answers
40 views

Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
7
votes
1answer
330 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
0
votes
1answer
50 views

Real Analysis limit

Well, let $f:\Bbb R\to \Bbb R $ a Riemann integrable function in $[0,2\pi]$, $2\pi$ periodic. I must prove that $$\lim_{n\to \infty} \frac {1}{2\pi} \int_{0}^{2\pi} f(x)g(nx)dx=\frac {1}{2\pi}\int_{0}^...
3
votes
1answer
92 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
2
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0answers
80 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
0
votes
1answer
55 views

Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
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0answers
56 views

A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
2
votes
2answers
38 views

Which function satisfies the conditions?

I'm solving a problem and, in order to run test case, I need a function $ b(x,y) $ that satisfies: $$ \int_0^L \int_0^H b(x,y) \, dx \, dy = 0 $$ and $$ \int_0^L \int_0^H b(x,y) \cos \left(\frac{n \...
0
votes
1answer
260 views

Fourier Transform of $\cos(\pi t)+2\sin(3\pi t)+\cos(5\pi t)\cos(7\pi t)$

I think I understand this question up until the last term. So far I have: $$F(\omega)=\pi(\delta(\omega+\pi) + \delta(\omega-\pi))+2\pi(\delta(\omega+3\pi) + \delta(\omega-3\pi))+\frac{1}{2}(\delta(\...
5
votes
4answers
230 views

find the sum $\sum_{k=1}^{\infty} \frac {\sin^2(kx)}{k^2}$, for $x\in [-\pi,\pi]$

find the sum $$\sum_{k=1}^{\infty} \frac {\sin^2(kx)}{k^2},$$ for $x\in [-\pi,\pi]$. Also i know that $$\sum_{k=1}^{\infty} \frac {\sin(kx)}{k}=\frac {\pi-x}{2}$$ Any help would be much appreciated.
1
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2answers
44 views

Problem with the Fourier transform of a function

I'm having some troubles with this one: $$\mathcal F(e^{-|x|} +|x|e^{-|x|})$$ I know that $\displaystyle\mathcal F(e^{-|x|})={1\over \pi (1+w^2)}$ but the second part is where I get stuck.
0
votes
2answers
91 views

Convert phasors to sinusoidal waveform

I'm trying to convert this phasor into a sinusoidal waveform. $$ j6e^{-j\pi/4} $$ Here's what I have so far: $$ 6\sin(\omega t-\pi/4) = 6\cos(\omega t-\pi/4 - \pi/2) = 6\cos(\omega t-3\pi/4) $$ ...
1
vote
1answer
101 views

Inhomogeneous diffusion equation and initial conditions inversion

While working on a physical diffusion process, I encountered the following Fokker-Planck equation $$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$ where $D(x) > ...
2
votes
1answer
254 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
2
votes
0answers
148 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : \...
1
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1answer
39 views

Does there exists $f\in A(\mathbb T)$ such that $||f||=r$ and $||\mathrm{e}^{if}||= \mathrm{e}^{r}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t)\, \mathrm{e}^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):...
0
votes
1answer
95 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
0
votes
1answer
24 views

Finding Fourier coefficients of functions that are defined as integrals with known Fourier coefficients?

Given a continuous periodic function f, with a period of $2\pi$, and Fourier coefficients that are $\hat f(n) = \frac{1}{1+n^2}$ , what are the Fourier coefficients of $g(x) = \int_0^xf(t)dt $? So ...