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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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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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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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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105 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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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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490 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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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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29 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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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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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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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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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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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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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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Number Theoretic Transform (NTT) example not working out

I'm reading up on the NTT, which is a generalisation of the DFT. I'm working in $\mathbb{F}_5$ with primitive root $w=2 \mod 5$. Suppose I want to compute the NTT of $x=(1,4)$. So far I have obtained: ...
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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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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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If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall x \...
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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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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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477 views

Computing Fast Fourier Transform in R

This is a math question with programming application. So I am trying to find 3 things given a certain function in the x domain when transformed into the spectral domain. 1) the Amplitude 2) The ...
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539 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
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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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455 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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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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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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Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, k\...
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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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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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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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Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function $f\...
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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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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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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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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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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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About the Fourier-Legendre series of $f(x)=e^{-x}$

So for the function $f(x) = \exp(-x)$ I have the formula for the coefficients of $$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$ which is(by using Rodrigues formula) $$a_n = \frac{2n+1}{2} \int_{-1}^{1}\...
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Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
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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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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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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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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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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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66 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: $f(x)=\frac{1}{\pi}\...