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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Wave Equation: Distribution which maximises Entropy

Given the wave equation: $\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)f(t,x)=0$ and expanding $f(t,x)$ through a Fourier Transform: $f(t,x)=\int d\omega dk F(\omega,k)e^...
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23 views

What will be power spectral density?

If the autocorrelation of a random process is given as $$r_x(k) = \begin{cases} 10 - \vert k \vert &\mbox{if } |k| < 10 \\ 0& \mbox{else.} \end{cases} $$ What will be the DTFT of this ...
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25 views

Example of fourier series diverges at least two points

I know continuous function $f$ such that fourier series of $f$ diverges at one point. But, I don't know continuous function such that fourier series of $f$ diverges at least two point.
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50 views

From FT deduce the value of $\int_0^{\infty} {\sin^4 x \over x^4} dx$

We have the function $$f\left(x\right)=\begin{cases} 1+x, & -1\leq x\leq0\\ 1-x, & 0\leq x\leq1\\ 0, & \textrm{otherwise} \end{cases} $$ we computed its Fourier transform: $$\hat f(t) = {...
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Discrete Fourier transform of $e^{i \alpha n} \Theta(n)$?

I am trying to understand the discrete Fourier transform $\mathcal{F}[s_n](\omega)=\sum_{-\infty}^{\infty} s_n e^{-in\omega} $ of $$s_n = e^{i \alpha n} \Theta(n),$$ with the Heaviside step function $\...
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Convolution and Fourier transform problem

I was struggling with this question, can use some help. given that $a\not=0$ $$f_a(x) =\frac{1}{x^2+a^2}$$ I'm trying to find k and c dependent on a and b $$(f_a ∗ f_b) (x) = kf_c(x) $$ I know ...
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30 views

The indefinite integral of an $L^1$ function has a convergent series of Fourier coefficients

Let $f\in L^1([0,2\pi])$ be a $2\pi$-periodic function with $\hat f(0)=0$ and $\hat f(\vert n\vert)=-\hat f(-\vert n\vert)\geq 0$. Define $F(t)=\int_0^t f(x)dx$. I know that F iscontinuous, $2\pi$-...
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73 views

What assumptions do we need for Fourier transform of derivative formula

Suppose $f: \mathbb R \to\mathbb C$ is differentiable and $f$ and $f'$ are in $L^1(\mathbb R)$. Do we need further assumptions to have the formula: $$\widehat{f'}(t) = (2\pi it)\hat f(t) $$ My ...
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Is it possible to conveniently express the probability density of a vector by its Fourier transform?

Let $x$ and $\hat{x}$ be a random vector and its Fourier transform, respectively. For any practical purpose we can assume it's a finite vector and the Fourier transform is given by the DFT, i.e. there ...
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28 views

Proof of Fourier Shift Theorem

The Fourier transform of $f(x)$ is, by definition, $$ \mathcal{F}[f(x)](u) = \int f(x)e^{-ixu}dx.$$ Now, the Fourier transform of $f(x-a)$ is, according to the Fourier shift theorem, $$ \mathcal{F}[f(...
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Why is the inverse fourier transform of $e^{-\omega^2+2i\omega}$ of the form $e^{(x-2)^2}$ and not of the form $e^{(x+2)^2}$?

So wolframalpha gives me this: wolfram's result for inverse fourier transform but I have this relation in mind: fourier transform rule I'm using and combining this with the shift theorem I would ...
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1answer
30 views

Evaluate integral of $2\pi$ periodic function's multiplication

Let f and g be $2\pi$ periodic Riemann integrable funtions. I want to evaluate $\lim_{n \to \infty} \frac{1}{2\pi} \int_0^{2\pi}f(x)g(nx)dx$ I think that this integral express by using fourier ...
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28 views

Doesn't the recursive Fast Fourier Transform violate f(-x) =/= f(x) for odd functions?

When you recursively split into $Y_{even}$ and $Y_{odd}$, from the second recursion onwards don't these sets have their even-ness and odd-ness violated? I.e., assume you are running the FFT algorithm ...
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260 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=...
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Homogeneous solution, Fourier mode, wavelength

Consider a system $$ \partial_t u=N(\sigma)u,~~u=(x,t)~~~(1) $$ where $N$ is a non-linear operator depending on some control parameter. Suppose that the system (1) admits a homogeneous ...
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122 views

Averaging transformation of a closed plane curve

Let's suppose we have a closed plane curve of some shape whose points are described by the single parametric equation $P(x(t), y(t))$ where $t$ is some increasing parameter (example circle) or by ...
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66 views

Why the range of time period of exponential Fourier series is different from other two types of Fourier series?

Trigonometric Fourier series is given as $$x(t)=\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T) \quad(1)$$ Polar FS is given as $$x(t)=...
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27 views

Decay of Fourier Coefficients

Everybody knows that the Fourier coefficients of an $L^1$ function converge to zero. For an $L^2$ function we can say much more: $(\hat f(k))_{k\in\mathbb Z}\in\ell^2$. Therefore, it is reasonable to ...
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58 views

Bound related to Schwartz space

If $u \in \mathcal{S}'(\mathbb{R}^n)$, then is there an integer $m \ge 0$ and $C>0$ such that for all $\phi \in \mathcal{S}(\mathbb{R}^n)$,$$|u(\phi)| \le C\|\phi\|_m,$$where$$\|\phi\|_m = \sum_{|\...
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36 views

Equidistributed problem about polynomial with irrational coefficient

This problem is from Stein, Fourier Analysis,Chapter 4,problem 2(d). Problem:Suppose that $P(x)=c_n x^n+……+c_0$ is a polynomial with real coefficients, where at least one of $c_1,……,c_n$ is ...
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42 views

Calculating a uniform-in-wavelength Fourier transform

If $f$ is frequency in Hertz and we assume $c = 1$, then $\lambda = 1/f$ and we have the representation$$s(t) = \int_{-\infty}^\infty \hat{s}(1/\lambda)\exp(i2\pi t/\lambda)1/\lambda^2\,d\lambda.$$How ...
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Relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring for dummies?

As the question title suggests, what is an explanation for dummies of the relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring?
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40 views

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$?

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$ I have a proof for the first case, under the assumption that $f$ is $C^1$ and real valued (also $1$ periodic) $\lVert ...
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29 views

Proof check: Isoperimetric inequality

My Proof: Let the arc-length parametrization of the curve be $\gamma(s) = \langle x(s),y(s)\rangle$. By Green's Theorem, the area $\mathcal{A}$ is $$ \mathcal{A} = \int_{0}^{2\pi} -y(s)x'(s) dx$$ ...
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36 views

The Fourier series of a continuously differentiable function converges to it pointwise

$S_N(f):=\sum\limits_{|n|\le N}\hat f_n\cdot e^{i2\pi nx}$ where $\hat f_n=\int_0^1f(x)e^{-i2\pi nx} dx$ and $f$ a $1$-periodic $C^1(\mathbb R,\mathbb C)$ function, then $S_N(f)$ converges pointwise ...
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58 views

If a fourier series converges to an elementary function, can I then find the closed form of this function?

Suppose that I am told that f(x) is some elementary function and that f(x) has the fourier series $\Sigma_{k=-\infty}^{\infty}c_ke^{ikx}$. By "elementary function" I mean: https://en.wikipedia.org/...
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31 views

Can Mathematica/WolframAlpha do a Fourier transform for f instead of ω?

When Mathematica/WolframAlpha calculates the Fourier Transform, it calculates it using the angular frequency. How do I make the Fourier transforms Mathematica/WolframAlpha to match the following table?...
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37 views

If $A(x)$ is 1-periodic and $\frac{A''(x)}{A(x)} = C$, then $C=-4\pi^2 n^2$?

This might be a trivial question but I forgot my differential equation. Anyway, I am trying to solve the heat equation on circle. Given that $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\...
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35 views

Derivative of Fourier transform with respect to intermediate variable

I am studying a system with a characteristic, say $\zeta$, that varies in 3D real space. I can use this characteristic to calculate the value of a second characteristic $\beta$. In other words, I have ...
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39 views

Fourier Analysis / Real Analysis

I have run into the following exercice from Grafakos' Classical and Modern Fourier Analysis: if $f$ is a function in $L_{1}(R)$, then one has to prove that $\int_{-\infty}^{\infty} f(x)dx$ = $\int_{-\...
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21 views

Fourier Convolution Inversion

Consider a Fourier convolution $f(x) = (g * h)(x)$, where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ...
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Suppose $\sum |A_n|^2 <\infty$, then $\sum r^{|n|} A_n e^{inx}$ converges uniformly?

Suppose $\sum_{-\infty}^\infty |A_n|^2$ converges. Show that for each $r\in (0,1)$, the series $\sum_{-\infty}^\infty r^{|n|} A_n e^{inx}$ converges uniformly in $x$. I know that the series $\sum_{-\...
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Result of a decay condition

Assuming that a function g is such that $ g(x) \leq C ( 1 + |x|)^{(-1 - \varepsilon)}$ for some $\varepsilon > 0$ , then how can we prove that $ \sum_{n = - \infty}^{n = + \infty} | g(x- k - \frac{...
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Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$

Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$. $f(w)$ is a Schwartz function. This is a part of the proof of Fourier inversion ...
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Integration on $R^d$ about Changing Coordinate

I have the formula $$\int_{\mathbb R^d} F(x)dx=\int_{S^{d-1}}\int_0^\infty F(r\gamma)r^{d-1}drd\sigma(\gamma)$$ Problem: Apply this to $F(x)=g(r)f(\gamma)$, where $x=r\gamma$, to prove that for ...
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1answer
73 views

If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges

I'm learning about Fourier series, specifically Cesàro summable sequences and series, and need help with the following problem: Show that if the series $\sum_{k=1}^{\infty} a_k$ is Cesàro ...
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If $f \in L^2(R)$, then $\hat{f}\in L^2(R)$?

The Fourier transform of $f$ is as follows: $$\hat{f}(\omega)=\frac{1}{‎‎‎\sqrt{2‎\pi‎}}\int_{-\infty}^{+\infty} e^{-i\omega t}f(t)dt$$ I need to know that if $f \in L^2(R)$, then can we conclude that ...
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If $f \in L^1(\mathbb{R})$, then $\hat{f}(\omega)‎\to0$ as $‎\omega‎ \to+‎\infty‎$?

The Fourier transform of $f$ is as follows: $$\hat{f}(\omega)=\frac{1}{‎‎‎\sqrt{2‎\pi‎}}\int_{-\infty}^{+\infty} e^{-i\omega t} f(t) \, dt.$$ I need to know that if $f \in L^1(R)$, then can we ...
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1answer
38 views

If $f: \mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic, then $f \in L^2[-\pi, \pi]$

I'm learning about Fourier series, specifically $L^2$ convergence, and need help with the following problem: Let $f: \mathbb R \to \mathbb R$ be continuous and $2\pi$-periodic. Show that $f \in L^...
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1answer
50 views

Fourier series of piecewise-defined function and convergence

I'm learning about Fourier series and need help with the following problem: Consider the function $$g(x) = \begin{cases} x^{\frac{1}{3}}, & x \in [0, \frac{\pi}{2}] \\ (-x)^{\frac{1}{4}}...
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1answer
31 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
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Proving, that $\text{Arg}(-i\sin(x))=\pi/2\text{sgn}(x)$ on $(-\pi,\pi)$

Alright. I thought, that $\text{Arg}(-i\sin(x))=3\pi/2$, however, the Wolfram Alpha tells a different story. I am sure that it must be kind of true, because $\text{Arg}(\sin(x))$ is the result of sum ...
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21 views

Relation of rate of decay of a function with width of peaks of its Fourier transform

Consider a function $f(t)=\theta(t)e^{-\sigma_0 t}\sin(\omega_0 t)$, where $\theta(t)$ is $1$ for positive $t$ and $0$ for negative $t$. Its Fourier transform can be easily computed. It has the ...
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Absolute maximum

I´m trying to find the absolute maximum of $(2N-1)$ partial sum of the Fourier´s series of signum function on $[0,\pi]$, I have: $S_{2N-1}[f](x)=\frac{4}{\pi}\displaystyle\sum_{k=0}^{N-1}{\frac{sen((...
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1answer
98 views

Parseval's identity holds

Theorem: If $u \in L^2(\mathbb{R}^n)$ then the Fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: $||\widehat{u}||_{L^2(\mathbb{R}...
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1answer
45 views

Can you recover a distribution from mollification?

Let $f\in \mathcal S'(\mathbb R)$ be a Schwartz distribution. Given $\rho \in C^\infty_c(\mathbb R)$ define the convolution as the function $$x\mapsto (f\ast\rho)(x):=\langle f, \rho (\cdot -x)\...
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2answers
40 views

Finding the coefficients of a triangular wave.

I have the following equation that I want to solve $$a_k = \color{blue}{\frac{1}{T} \int_{0}^{T/2} 2 \frac{t}{T} e^{-i \frac{2\pi}{T}kt} dt} + \color{red}{\frac{1}{T} \int_{T/2}^{T} 2 \frac{T-t}{T} e^...
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54 views

Relation between Dirac function and inverse fourier transform of 1

According to my notes, it holds that $\delta=(2 \pi)^{-n} \widehat{1}$. How do we get the equality? We have that $\delta=\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{\delta}(\xi) e^{i x \xi} d{\...
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1answer
38 views

calculate Fourier Transformate

i have the following exercice: Let for all $x \in \mathbb{R},$ $f(x)= \cos x$ and $g(x)= \sin x$. Calculate $T=f \delta' + g \delta''$ for this question, i find $T=3 \delta$. Calculate the Fourier ...
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2answers
46 views

Is the Hilbert transform of a Schwarz function essentially bounded?

My ultimate goal is to show that the Hilbert transform of a Schwarz function is in $L^p(\mathbb{R})$, for every $p \in (1,\infty]$ (the definition I am using is $Hf(\xi) := \mathcal{F}^{-1}[(-i \ \...