# Tagged Questions

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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### 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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### 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 $$f(x)=1+x^3$$ and the definition of Fourier transform: \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, k\...
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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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### 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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### 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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### Difference between Fourier integral and Fourier transform

What is the difference between Fourier integral and Fourier transform? I know that for Fourier integral the function must satisfy that : $\displaystyle \int_{-\infty}^\infty |f(t)| dt < \infty$, ...
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### Fourier basis functions

What are Fourier basis functions? And how do I prove that Fourier basis functions are orthonormal?
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### Wiener filter: A good tutorial

I am interested in image analysis and am looking for an approachable tutorial to the Wiener filter. At some point I am interested in implementing such a filter but I would like to have a deeper ...
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### 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 ...