# Questions tagged [fourier-analysis]

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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### Discrete Fourier transform of $(1,1,1,1)$

I am asked to determine the Fourier transform of $(1,1,1,1)$. In the solution I found this: I don't get how is he transitioning from the $\omega$'s to $-i, i, -1, 1$ etc... How to break it down, so ...
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### Discrete Fourrier transform

I'm learning the DFT (Discrete Fourrier Transform), and came across a mathematical sign, that I didn't understand. They say (Inverse of DFT Matrix) F^-1 * c = v And then, there is an equality that ...
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### A piecewise function as the output signal of an LTI system

In an LTI system, consider the following: The input signal: $$x(t)= \begin{cases} 16 \quad & ; -7<t<0 \\ 0 & ; \text{otherwise} \\ \end{cases}$$ And the unit impulse ...
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### Peter-Weyl Theorem on the Sphere

The Peter-Weyl theorem says that the matrix coefficients of the unitary irreps of a compact topological group $G$ form an orthonormal basis for $L^2(G)$. Similarly, spherical harmonics provide an ...
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### Orthonormal basis for $\mathcal{L}^2([0,1])$

$\textbf{Theorem:}$ The orthonormal family $\{e_n(x):\, n\in\mathbb{N}\}$, where $e_n(x)=e^{2\pi inx}$, is a basis for $\mathcal{L}^2([0,1])$. In this case, $\{e_n(x):\, n\in\mathbb{N}\}$ being a ...
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### Fourier Transform An FM synthesized wave.

Greetings StackExchange, (I hope this question comes under Mathematics and not Physics) I've been attempting a Fourier Transform on an FM synthesized wave (as below). After a long time (about 8 days) ...
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### how to find the discrete time fourier series of sin function

I have this discrete signal sin($0.1 \cdot \pi n$) and I want to find the DTFS of it. first I have found its fundamental period N which is 20 but what am struggling with is how am I going to find 29 ...
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### The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
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### Can someone help me with Fourier? [on hold]

can someone give me a hand with the section c) and d)? I can not do it ... I think the point is that I do not understand how to integrate the function and then draw what it asks for. Problem
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### Conceptual Doubt regarding principle of superposition for Linear PDE(s)

This is in regards to a problem I have been trying to solve. I have already posted a question regarding it previously, but the part of the problem I describe here leans more towards conceptual ...
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### Fourier transforms and the angular frequency

In the context of Fourier transforms, is the following true? $$\mathcal{F} \left( F(t) \right) =2\pi \, f(-\omega )$$ If it is not, what small changes can I make to the equation to make it true?...
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### oscillatory integrals in several variables with nonstationary pahse

Consider the oscillatory integral $$I(\lambda):=\int_{\mathbb{R}^{n}} \mathrm{e}^{\lambda\dot{\imath}\phi{(x)}} \psi{(x)}dx$$ where the phase $\phi$ and the amplitude $\psi$ are smooth real-valued ...
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### While Fourier series solve heat equation on a finite interval, can Fourier transform solve heat equation on infinite line?

Consider the heat equation with initial value $f$ $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\ \ \ u(x,0)=f(x).$$ We look at two cases, one with $x\in[-\pi,\pi]$ (heat equation ...
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### Can we bound the value $\int_{[-A,A]^c} | \hat f(k)| dk$ where $A>0$ and $\hat f$ is the Fourier transform?

Assume $f \in L^1$ and $\hat f \in L^1$. Then it is easy to see that for $A>0$ the value $$D := \int_{[-A,A]^c} | \hat f(k)| dk$$ tends to zero for big $A$ ($[-A,A]^c$ is the complement of the ...
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### $f$ is continuous on the circle, and its Fourier series is absolutely convergent. Then its Fourier series converges uniformly to $f$.

This Corollary's proof (2.3 in Stein's book), is not sufficiently detailed for me to fully understand. I can see why the Fourier series uniformly converges to some continuous $g$ but how can I ...
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