# 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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### Approximating Fourier transform for range of output frequencies

(This may be an elementary question, I am new to Fourier analysis.) I am working on a visualization tool. I have a real function $f(x)$, given by N samples on some interval, and vanishing outside ...
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### Recommend resources for understanding Phase spectrum

I am learning Fourier transform. if we apply Fourier transform on a signal, we get magnitude spectrum and phase spectrum. I want to learn phase spectrum part in detail. So can anyone recommend any ...
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### Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right)$$ ...
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### Find the Fourier transform of $\sin x^2$.

I've tried it by applying integratrion by parts, but I'm not getting the answer correct. Its answer is $$\frac{1}{\sqrt{2}}\,\sin\left(\frac{k^2}{4} +\frac{\pi}{4}\right).$$ Please help in this.
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### How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
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### Fourier transform of a continuous periodic spectrum of frequencies

Suppose I have a function of the form $$f(t) = \exp(i\phi(t))$$ where $$\phi(t) = \int_0^t\omega(t) \ dt + \phi_0$$ is the phase of the function and $\omega$ is the angular frequency, which is ...
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### Théorie de Fourier in Sontags book

I was reading Sontags In America and she mentions: "La théorie de Fourier sur les douze passions radicales.." What is this theorem about?
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### Evaluate $\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$ where $g(x) = \int_{0}^x f(t) dt$

Let $f$ be a $2\pi$-periodic function such that $\int_{-\pi}^\pi f(t) dt = 0$. Define $g(x) = \int_{0}^x f(t) dt$. Evaluate $$\frac{1}{2\pi}\int_{-\pi}^\pi g(x) dx$$ I hope the integral is equal ...
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### Prove that $\mathscr{F}[f] \in L^2(\mathbb{R})$

Let $f \in L^2(\mathbb{R})$ (square integrable functions), I'm trying to prove that his Fourier transform also does: $\mathscr{F}[f] \in L^2(\mathbb{R})$. I have tried to bound it \begin{align} \...
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### How describe functions with finite bandwidth?

What is the sufficient and necessary conditions for a $f:\mathbb R\to\mathbb R$ has finite bandwidth (Fourier spectrum is non-zero on a bounded interval)? I'm guess this equivalent $f$ is continuous ...
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### Computing Hilbert transform and envelope of a function

The following is a function with $\alpha$ being a real constant $$f(t) = \frac{\sin(\alpha t)}{\alpha t}.$$ Determine the analytic signal $f_a (t),$ Hilbert transform $\hat{f}(t),$ and the envelope ...
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### If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?
if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
### $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$
I am trying to establish the following $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$ The right sight looks the the Fourier expansion of ...