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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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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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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2answers
18 views

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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1answer
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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1answer
30 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
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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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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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44 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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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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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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25 views

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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1answer
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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22 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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27 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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1answer
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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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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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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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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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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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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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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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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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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17 views

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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1answer
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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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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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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1answer
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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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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31 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 ...
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27 views

How to solve $u_{tt}+\Delta u + x^{2}u=0$?

Let $u:\mathbb R \times \mathbb R \to \mathbb C$ be some function so the everything in the following make sense. Consider the following PDE: $\frac{\partial^{2}}{\partial t^2} u(x,t) + ...
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23 views

A $H^p$ function

Set $\mathbb U=\{x+iy|\;y>0\}$. A function $f:\mathbb U\to\mathbb C$ is called a $H^p$ function if $f(z)$ is holomorphic and $\|f\|_{H^p}:=\sup_{y>0} \left(\int_{-\infty}^{\infty} |f(x+iy)|^p ...
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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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1answer
20 views

Relation between Fourier Transform Duality and other properties.

I'm having a hard time with Fourier Transform's Duality Property. The Duality Property states that, if $$\mathcal{F}\left\{x(t)\right\} = X(\nu),$$ then $$\mathcal{F}\left\{X(t)\right\} = 2\pi ...
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1answer
32 views

Asymptotic behavior of Fourier transform and Hölder condition

I'm trying to solve this question. Following the hint, the Fourier inversion formula gives me : $$ \big| f(x+h) - f(x)\big| = \left| \frac{1}{2\pi} \int_{-\infty}^\infty \widehat{f}(\xi) ...
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50 views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
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Fourier coefficients of a discontinuous function

For the following given function $J$ It is not continuous but nonetheless integrable. My professor mentioned that it is still possible to "find a Fourier series corresponding to $J$" and his proof ...
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33 views

Computing the complex fourier series for triangular wave from trigonometric coefficients

I'm trying to figure out how to compute the complex Fourier series for the triangular wave, given the trigonometric coefficients. The book gives as a result for the complex series the following: $$ ...
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1answer
33 views

laplace transform of $t^nf(t)$

I have: $$\mathcal{L}(t^nf(t)) = \int_0^\infty t^nf(t)e^{-st}\ dt = \left(-\dfrac{d}{ds}\right)^n \int_0^\infty f(t) e^{-st}$$ I don't understand where the derivative came from
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24 views

If a sequence of function converges in $L^2$ sense, then its Fourier series converges in $l^2$ sense?

I have the following material from my lecture notes and I am trying to prove it, but I am not sure how to do the second part. Suppose we have a sequence of continuous functions $\{q_n (x)\}$ ...
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1answer
23 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) ...
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show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] ...
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29 views

Show that f is constant by using fourier coefficient

Let f be a 2$\pi$ periodic, Riemann integrable function and let $\alpha$ be an irrational number. Suppose that $f(x+2\pi\alpha)=f(x)$ for all x. Show that f is constant almost everywhere. I know that ...
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23 views

How to prove the absolute value of this Fourier coefficient is bounded?

The question is: Let $f:[0,1)\to C $ be a step function. Prove that there exists a constant C such that for all integers $k\neq 0$, $$ \mid \,\hat{f}(k)\mid\le C\,\big/\mid k \,\mid.$$( $\hat{f}(k)$ ...
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1answer
51 views

What does the overline symbol mean?

So I have got a question in an old exam paper for Fourier Analysis. Let $f:I\to C$ be an integrable function. Prove that$\int_I \overline{f(x)}= \overline{\int_I f(x)}$. The problem is that I ...
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2answers
31 views

How to prove the Fourier coefficient of a convergent sequence converges as well

So the question is, $\,f_1, f_2,...,f:T\to C$ are integrable functions with $ \,f_n\to f$ in $\parallel \cdot \parallel_1 $ as $n\to \infty$. Let $k\in Z$. Prove that, the Fourier coefficient, $ ...