# 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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### What is the value of $\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{k=1}^\infty\Big(1-p+p\exp \frac{it}{2^k}\Big)dt$?

Here $0<p<1$. If $p=\frac{1}{2}$ the value is $1$. In all cases the value is a positive real number. This integral is associated with the inverse Fourier transform of some characteristic ...
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### How do I find the magnitude and phase of the frequency response function for a third order system using transfer functions?

A third order system is described by: $$\frac{{\rm d}^3 y}{{\rm d}t^3} + 2 \frac{{\rm d}^2 y}{{\rm d}t^2} + 6 \frac{{\rm d} y}{{\rm d}t} + 5y = u + 2 \frac{{\rm d} u}{{\rm d}t}.$$ Determine the ...
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### Find a certain function of moderate decrease

I am trying to solve the following problem: Find a function $f$ of moderate decrease such that $$\int_{-1}^{1} f(x-t) dt = e^{-2\pi |x-1|} - e^{-2\pi |x+1|} .$$ The first idea I had was to use the ...
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### Elliptic Fourier fit coefficients

I am trying implement a function that could fit an elliptic fourier curve on a set of border points of a detected object. I am using cv2.findContours to acquire border points from a binary image. Next ...
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### Uniform convergence of fourier series of step function [closed]

Consider the series $f(x) = \sum_{n=1}^{\infty} a_n \sin nx$, where \begin{align} a_n = \begin{cases} \frac{4}{n \pi}, &n \text{ odd} \\ 0, &n \text{ even} \end{cases}. \end{align} It is ...
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### Fourier representation for $\tan(x)$

Q: Which Fourier representation is suitable for $f(x) = \tan(x)$: Fourier trigonometric series, Fourier half-range expansion, or Fourier integral and why? Well I searched and found that: $\tan(x)$ ...
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### Is there a Mellin transform or an analogue on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$?

From Wikipedia, the Mellin transform is an isometry $M : L^2(\mathbb{R}^+) \mapsto L^2(\mathbb{R})$, $$\{M f\} (s) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}^+} x^{-1/2 + \mathrm{i} s} f(x) dx.$$ ...
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### Fourier analysis notation - Sh and Ch

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!
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### Fourier Transform of sinc function (confused about $\operatorname{sinc}(\pi x)$ and $\operatorname{sinc}(x)$)

I am confused about the fourier transform of the $\operatorname{sinc}$ function. First I don't know if $$\operatorname{sinc} (x) = \frac{\sin(\pi x)}{(\pi x)}$$ or \operatorname{sinc} (\pi x) = ...
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### Carleson-Hunt Theorem on $\Bbb R$.

Wikipedia states the following: If $p\in(1,\infty)$ and $f\in L^p(\Bbb R)$ then $f(x)=\lim_{A\to\infty}\int_{|\xi|<A}\hat f(\xi)e^{i\xi x}\,d\xi$ almost everywhere. This seems clearly ...
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### Twice continuously differentiable wavelets with compact support

Do there exists wavelets such that both mother wavelet and the scaling function have compact support and they are both twice continuously differentiable?
### Fourier Series of Brownian Motion $(B_t)_{t \in [0,1]}$ wrong?
In our lecture about Brownian motion, we calculated the fourier series of Brownian motion $(B_t)_{t \in [0,1]}$. We did the following: We first defined the Brownian bridge $(\beta_t)_{t \in [0,1]}$ ...
I'm trying to do the following exercise from Rudin's Real and Complex Analysis: Suppose $f\in C(T)$ and $f\in \text{Lip }\alpha$ for some $\alpha>0$. Prove that the Fourier series of $f$ ...