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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Find the function $f(x)$ by using its fourier coefficient

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem How to find the function $f(x)$, if I know its fourier coefficient (or ...
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19 views

Class of Functions , that admit Fourier transforms

For which class of functions/distributions is it sensible to take a Fourier transform ?
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43 views

Convolution of Schwartz and test function approximated by partition of unity.

Let $\rho\in\mathscr{D}$, $0\leq\rho\leq 1, \rho(0) = 1$, and $\sum_{n\in\mathbb{Z}^d}$ $\rho(x-n) = 1$. Denote, $\rho_{n,\epsilon}() = \tau_n\rho(\frac{x}{\epsilon})$, where $\tau$ is the translation ...
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50 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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24 views

Fourier Series and Fourier Transform confusion.

I dont understand the following paragraph after the proof. In particular, how does that theorem above give us that the Fourier transform maps $L^2$ onto $l^2$? all that theorem says is that this set ...
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Fourier Transform leading to $\delta$: How does the Integration work?

So it is well-known that the complex exponential $$f(t) = e^{i\omega_0t}$$ has Fourier transform $$F(\omega) = 2\pi \delta(\omega-\omega_0) \ .$$ The transformation integral $$F(\omega) = ...
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7 views

Deconvolution by disks

I have a function $$f(x,y) = \begin{cases} 1, \|(x,y)-p\| < r\\0, \|(x,y)-p\|\geq r\end{cases}$$ where $p$ is some unknown point in $[0,1]^2$; i.e. $f$ is the characteristic function of some disk ...
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Gaussian is a rapidly decreasing function.

Definition of rapidly decreasing function $$\sup_{x\in\mathbb{R}} |x|^k |f^{(l)}(x)| < \infty$$ for every $k,l\ge 0$. Given the Gaussian function $f(x) = e^{-x^2}$, I know that its derivatives ...
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28 views

Dirac function expansion

In my book it is said that Dirac function $\delta(\tau)$ can be expanded as: $$ \delta(\tau)=(\beta \hbar)^{-1}\sum_{n \in even} e^{-i\omega_n\tau} $$ where $\omega_n=\frac{n\pi}{\beta\hbar}$, and ...
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29 views

How to get from $\sum_{n=0}^N (a_n \cos{nx} + b_n \sin{nx})$ to $\sum_{-N}^{N} c_n e^{inx}$?

I'm currently reading Baby Rudin, and I'm in the section of Chapter 8 that covers Fourier series. There is one line that I just can't figure out for the life of me, and I can't find anything online ...
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13 views

What if we define function of moderate decrease as function satisfying $|f|\le\frac{A}{x^2}$

In Stein's Fourier Analysis, he defines the a continuous function $f$ as of moderate decrease if there exists $A>0$ such that $$|f(x)| \le \frac{A}{1+x^2} \forall x\in\mathbb{R}$$ I am wondering ...
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25 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ ...
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42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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10 views

upsampling and plotting a signal in matlab

I want to upsample by 5 a signal in frequency domain, and then plot(stem) it. I figured how to upsample, Fk=(1/5)*upsample(ak_new,5) now this creates a vector ...
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19 views

Uniform convergence of Fourier series given certain conditions

If $f$ is a continuous, $a$-periodic and piecewise differentiable function on $[0,a]$ with piecewise continuous derivative on $[0,a]$, then $(f_N)$ converges uniformly to $f$ over $\Bbb R$. ...
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29 views

Coercitivity of an elliptic operator with constant coefficients

We are given an elliptic operator $P=\sum_{|\alpha|\leq m}a_\alpha\partial^\alpha$ that is elliptic in $\Omega$. $a_\alpha$ are constants. I am supposed to show that $$\|u\|_s\leq ...
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14 views

Prove that if $f \in C^r(T)$, then $\hat{f}(n) = o(\frac{1}{|n|^r} )$ as $n \rightarrow ±∞$

I searched through everything that came up when I searched this question, but didn't come with anything. I'm used to typing in latex, so please excuse any formatting errors.
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1answer
24 views

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$. I am not quite sure how to start this problem. ...
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992 views

Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where ...
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393 views

About the order of the $L^1$ norm of the Dirichlet kernel.

Reading this text from Wikipedia, I found the following statement about the Dirichlet kernel: $$\| D_n \|_{L^1} \approx \log n, $$ where $\approx$ denotes "is of the order". I think that this mean ...
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How do I calculate the Fourier Transform of this signal?

The Context: Find $X(ω)$ which is the frequency domain representations of $x(t)$. $$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$ This my professor's solution: As we can see, the ...
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21 views

Use trigonometric polynomial to approximate periodic function.

From page 53 of Fourier Analysis by Stein, we have If $f$ is integrable on the circle, then the Fourier series of $f$ is Cesaro summable to $f$ at every point of continuity of $f$. Moreover, ...
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45 views

Contradiction between Fourier and Laplace transforms?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has both Fourier and Laplace transforms. Also let $f(t)=0$ for all $t<0$. The Fourier transform of $f$ is ...
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76 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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27 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
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1answer
55 views

Fourier transform of isotropic Laplace distribution (2D)

How would I evaluate the Fourier transform of an isotropic 2D Laplace distribution? $F(\omega_x,\omega_y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-b \sqrt{x^2+y^2})\exp(-j\omega_x ...
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25 views

Fourier series in spherical coordinates?

I'm reading an article and he just state: let $f\left(\theta,\varphi\right)$ be of this form $$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},$$ I'm on the unitary ...
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68 views

Given that $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, how can I find $\sum\frac{1}{n^6}$?

Given that $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, how can I find $\sum\frac{1}{n^6}$? I know that you can prove the first equality using Fourier analysis. For the second one, do I try to use a ...
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32 views

Determing an inverse Fourier transform

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to ...
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Why we need fourier series to analysis wave spectrum?

I am interested to study water wave spectrum to analysis irregular wave height. To analysis wave spectrum we need fourier series but i have a question to mine why we need to study fourier series.
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454 views

Using partial fraction decomposition to find inverse Fourier transform

I've reduced my problem to $H(w) = \dfrac{1}{(1-\frac{1}{4}e^{-jw})(1-\frac{1}{3}e^{-jw})}$. I need its inverse discrete Fourier transform. My thinking is that I could use partial fraction ...
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1answer
14 views

inverse fourier transform of w*e^w

I have the function \begin{align} F^{-1}\{{λe^{-|λ|}}\} \end{align} How can we find the inverse Fourier transform? The correct answer is: \begin{align} \frac{-2ix}{π(1+x^2)^2} \end{align} Can ...
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Solving Viscous Burgers using spectral method

I am trying to solve the Viscous Burgers equation using the spectral method. $$u_t+uu_x = Du_{xx}$$ where $D$ is a constant (chosen to be zero) and with the initial condition $$u(x,0) = ...
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29 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of ...
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Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the $n$th order partial derivative of a function with respect to $x$, using the transform variable $k$ is: $$(i*k)^n * [F(k)]$$ so for the $1$st order ...
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58 views

Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...
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1answer
16 views

Can we relax the hypothesis of Uniqueness theorem for Fourier series?

I know this fact: "Suppose that $f\in L^{1}(\mathbb T)$ and $\hat{f}(n)=0$ for all $n\in \mathbb Z,$ then $f=0 $ all most everywhere on $\mathbb T$." My Question is: Suppose that $f\in ...
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Question about Dirichlet kernel of Fourier transform for $f\in L^p$ with $p\in [1,2]$, help needed in understanding proof.

I am trying to understand the proof that the following two statements are equivalent. For fixed $R>0$ and $f\in L^p(\mathbb{R}^n)$ let $$S_Rf(x)=\int_{|\xi|<R} \hat{f}(\xi)e^{2\pi i x . ...
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How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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13 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
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Is $A(D)$ a complemented subspace of $C(T)$?

Let $T$ be the unit circle and $D$ the open unit disk. A function $f$ belongs to $C(T)$ if it is continuous at $T$. A function $g$ belongs to $A(D)$ if it is continuous at $\overline{D}$ and ...
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Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$ A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
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How can I convince myself of the Fourier scaling property via inverse FT?

I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$: \begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= ...
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33 views

Fourier transform of product of twho functions that includes characteristic function

I need to find the fourier transform of f(x)=(1-abs(x))(Chi_[-1,1] (x)). In words, I need the fourier transform of one minus the absolute value of x multiplied by the characteristic function on ...
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18 views

What effect does sampling time have on a Fourier Series sum?

What effect would the sampling time of this Fourier sum have on it's accuracy? Is this to do with Nyquists theorem? or am I heading in the wrong direction with this question? Cheers
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42 views

Sobolev and Fourier

If we have $f=1_{[a,b]} \varphi$ with $\varphi \in \mathcal{D}(\mathbb{R})$, we found that the sufficient and necessary conditions to have $f\in H^1$ is that $\varphi(a)= \varphi(b)=0$. If we take ...
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68 views

Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.

Theorem 2.1 : Suppose that $f$ is an integrable function on the circle with $\hat f(n)=0$ for all $n \in \Bbb Z$. Then $f(\theta_0)=0$ whenever $f$ is continuous at the point $\theta_0$. Proof ...
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1answer
686 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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47 views

What am I doing wrong when I try to deduce the Laplace transform formula?

The Laplace transform of a function $f(t)$ is the projection of $f(t)$ vector (indexed with $t$) onto the linearly independent set of vectors $e^{st}$. The projection of a vector $\vec{v}$ onto ...
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30 views

Fourier sine and cosine series: reconstruction is shifted with respect to measured data

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...