# 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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### How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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### Looking for guidance on a Fourier integral

Working with a Fourier transform problem, I've encountered the following integral: $$\int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx$$ where $a$, $b$, and $c$ are real ...
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### Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $$y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$$ I am having some ...
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### Uniform convergence of Fourier Series

I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand: Given that we know the series $f(x) = \sum c_k e^{ikx}$ ...
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### $L_{p}$ distance between a function and its translation

I'm working through a proof and one of the comments is that for a function $f\in L_p (\mathbb{T})$: $$\lim_{t\to 0}\;\|f(\cdot + t) - f\|_p = 0.$$ How do I prove it? I think it is intuitively ...
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### Is the Fourier series always the “best” approximation?

In Stein and Sharkachi's Introductory Fourier Analysis book, in Chapter 3, we're given the "Best Approximation Theorem". The theorem says that if $f$ is Riemann integrable on the circle and we ...
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### Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$

Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \...
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### Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
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### Two definitions of Fourier transform for $L^1$ and $L^2$ coincide

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
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### What are the properties of the fourier transform of a phase-only function?

Given a function of the form: $$f(x) = e^{i\phi(x)} | \phi(x)\in\Re$$ What are the properties of its Fourier transform? For instance, purely real functions have Fourier transforms with symmetric ...
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### Closed epicycloids

Consider a system of $n+1$ celestial bodies - e.g. a sun, a planet, a moon, a satellite of the moon and a satellite of the satellite - which run around each other on circles in the same plane but with ...