# 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 ...
Show the Fourier transform $\mathcal F$ is continuous in the Schwartz space $\mathcal S(\Bbb R)$. Use the standard $\mathcal S$-norms $$\|f\|_{a,b}=\sup_{x \in \Bbb R} \left| x^af^{(b)}(x)\right|, \, ... 1answer 760 views ### Exercise 22, Chapter 5 of Stein and Shakarchi's Fourier Analysis I am working through Stein and Shakarchi's Fourier Analysis and am stuck on Exercise 22 of Chapter 5, which I quote below. Preliminary notation: \mathcal{S} is the Schwartz space of functions on \... 2answers 494 views ### Eigenfunction of the Fourier transform I want to show that$$ \frac{1}{ \sqrt{ 2 \pi}} \int_{-\infty}^{\infty} \frac{e^{-iwx}}{\cosh{ (x \sqrt{\frac{\pi}{2}}} ) } = \frac{1}{\cosh{ (w \sqrt{\frac{\pi}{2}}} ) } .$$My attempt is to ... 3answers 290 views ### Is there a name for a function whose square is an involution? An involution is a function f:X\to X such that f\circ f=\text{id}. Is there a name for a function g:X\to X such that f\equiv g\circ g is an involution? An example is multiplication by \pm i ... 2answers 2k views ### Hilbert transform and Fourier transform Assume the following relationship between the Hilbert and Fourier transforms:$$ \mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)), $$where  \displaystyle [\mathcal{H}... 3answers 272 views ### Fourier transform of e^{-i\lambda\sqrt{1+x^2}} - asymptotics for \lambda? As the title says: I want to compute the Fourier transform (in the distributional sense) of f(x)=e^{-i\sqrt{1+x^2}}, x\in \mathbb{R}^n - say n=1 for the moment. I have no idea how to get it done:... 1answer 220 views ### Is there a continuous waveform that sounds the same as a square wave? The fourier series$$f(t)=\sum_{n\in\mathbb N\\n\text{ odd}}\frac1n\,\sin(nt) converges to a square wave. Square waves are discontinuous functions. I'm wondering if there's a continuous function ...
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 ...