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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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Ηow do I find the Fourier transform of the cardinal sine?

Find the Fourier transform of $$\dfrac{\sin(a t)}{t\pi}$$ I tried using the formula but I can't get it to work, I asked my teacher and he said to use the proprietary of duality, but I don't ...
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Calculate the Fourier transform using the Airy function [on hold]

Calculate the Fourier transform from (using the Airy function): $$ \Large e^{2ix^3} $$
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Showing $||f - S_{n}(f)||_{2}^{2}$ $=$ $||f||_{2}^{2}$ $-$ $||S_{n}(f)||_{2}^{2}$

I am learning about Fourier Series in Carother's Real Analysis. We have just learned the Fourier Partial Sum $S_{n}(f)$ is the closest function to $f$ in the set of trigonometric polynomials with at ...
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0answers
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Convergence of Hermitian inner product in $l^2(\mathbb{Z})$

In the vector space $\ell^2(\mathbb{Z})$ over $\mathbb{C}$ (i.e. the set of all two-sided infinite sequence of complex numbers such that $\sum_{n \in \mathbb{Z}} |a_n^2| < \infty$) why is it ...
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3answers
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Why are the limits of the fourier cosine/sine series [0,∞) while the limits of the fourier exponential series are (-∞,∞)?

Ok so bear with me here. I am a computer/electrical engineering student who somehow got sucked into math land. Recently, I have been examining the Fourier series and it seems that there are two ...
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1answer
23 views

Sign convention for Fourier transform and contour integration - example

I was wondering about one (probably trivial) fact during computing the Fourier transform while using contour integral. As an example I have following function: $$f(x)={{1}\over{x^2+a^2}}$$ and its ...
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Explicit wave-package solution of the Klein Gordon equation

I need to generate initial for a 3D Klein Gordon (KG) solver. Therefore I'm interested in physical meaningful wave-package solutions. By considering the free KG equation \begin{equation} \Box \psi(t,x)...
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1answer
34 views

Why can you replace $1/\sin(\phi/2)$ with $2/\phi$ in an integral?

I am walking myself through a proof of convergence of Fourier series. For the partial sum $S_Nf(\theta)$, and any constant $S$, we have that $$ S_Nf(\theta) - S = \frac{1}{2 \pi} \int_0^\pi ( f(\...
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30 views

What is the minimal requirement for a Fourier transformation?

I have a lot of comprehension questions that I can't really figure out by googling: The difference between a Fourier transformation and a Fourier series. Am I correct in thinking that the difference:...
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1answer
26 views

$f(x) = \sum_{n=1}^\infty \sin (n x) / n^2 $ not differentiable at $x=0 $.

Apparently, by Weierstrass' test, this function is continuous. How to prove that it is not differentiable at $x=0$? I plotted its graph. It seems that $$\lim_{x\rightarrow 0} \frac{f}{x} =\infty . $$...
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1answer
25 views

Show this function is a good kernel on unit disk

I am having a bit of trouble showing the following function is a good kernel on the unit disk: $$ U_r(e^{i \theta}):=\frac{(1+r)^2(1-r)\theta\sin\theta}{(1-2r\cos\theta + r^2)^2}, \text{for}~~ 0<r&...
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How can I solve Fourier series? [on hold]

I've defined a function, but how do I decide? Condition of the problem: to Determine the values of the imaginary part of the Fourier coefficients Im{Cn}, n != 0 (not equal to), next sequence: Click ...
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0answers
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Fourier transform of a random walk

I was just reading Statistical Field Theory of Itzykson and Drouffe and saw that they wrote the inverse Fourier transform $$P(\vec{x},t,\vec{x}_0,t_0)=\int_{[-\pi,\pi]^d}\frac{\text{d}^d\vec{k}}{(2\pi)...
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1answer
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Converse of Fejer's Theorem

I am taking a course in Fourier Analysis this year. One of the theorems my lecturer wrote down was the following: Let $f \in \mathcal{L}^{1}([-\pi,\pi])$ be $2\pi$-periodic. Then, $f \in \mathcal{C}...
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1answer
35 views

An identity for Fourier transform of measure

Consider a finite Borel measure $\mu$ on $\mathbb R$. The Fourier transform $\hat{\mu}$ of $\mu$ is defined by $\hat{\mu} (\xi)= \int _{\mathbb R} e^{-ix\xi} d\mu(x)$. I would like to prove the ...
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Coefficients of the Fourier-Bessel Series if $f(r,\theta)$

This is directly related to my last post General Solution for $u(r,\theta, t)$. I have found that the general solution for $u(r,\theta,t)$ is, $$\sum_{k=1}^{\infty}\sum_{m=1}^{\infty} C_ke^{-(\mu^n_m)...
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Fourier transform of $\mathrm{e}^{-x^2 + i x^4}$

I would like to know if there is an analytical solution to the Fourier transform of $$\mathrm{e}^{-x^2 + i x^4}\,,$$ where $i$ is the imaginary unit. More generally, is there an analytical solution ...
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1answer
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For references on termwise differentiation of Fourier series?

I have seen a result on termwise differentiation of Fourier series: If$ f $ is periodic with period $ 2l $, continuous on $ \mathbb{R} $ and $ f' $ is piecewise continuous on $ [−l,l] $, then the ...
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1answer
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Proving continuity and boundedness of the Fourier Transform

Let $f\in L^{1}(\mathbb R^d)$. Define the Fourier transform of $f$ by $$\hat{f}(y)=\int_{\mathbb R^d}f(x)e^{-2\pi ix\cdot y}\,dx\,\,\,\,(y\in\mathbb R^d).$$ Show that $\hat{f}:\mathbb R^d\to\mathbb{...
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What is the Fourier transform of $\: e^{at}u(t)$? [closed]

I have been trying to find this answer, I have searched all over google, but I can't find the answer. Please help. Edit: Apparently my question has not been clear to many, so basically what I am ...
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0answers
19 views

Question about roots of unity in the Fast Fourier Transform

I am learning about the Fast Fourier Transform, which converts a polynomial from its coefficient representation into its point-wise form using divide-and-conquer. The Fast Fourier Transform evaluates ...
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2answers
28 views

Fourier transform of function of order $e^{-x^2}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a measurable function such that $|f(x)|=O(e^{-x^2})$ as $|x|\to\infty$. Does it imply that $\hat{f}\in L^1(\mathbb{R})$? ($\hat{f}$ is the Fourier transform of $f$.) ...
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Fourier spectrum of a particular boolean function

I'm trying to compute the Fourier spectrum of the following boolean function, $f_q :\{0, 1\}^n \to \{\pm 1\}$, where $f_q(x) = (-1)^{q(x)}$. Above, I pick $q(x) = x_1x_2 + x_2 x_3 + \cdots + x_{n-1} ...
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1answer
51 views

Fourier coefficients of $|\cos x|$

I’d like to find the Fourier coefficients of $$ x(t) = |A \cos{ ( 2 \pi f_0 t )} | $$ I found that, without taking the absolute value, the coefficient for $k=1$ is $\frac A2$ but now I don’t know ...
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Show ${na_n},{nb_n}$ are bounded [closed]

Let $\frac{a_0}{2}+\sum_{n=1}^\infty(a_ncos(nx)+b_nsin(nx)$ be the fourier series of a function $f\in BV[-\pi,\pi]$. Show that ${na_n},{nb_n}$ are bounded sequences. I am not sure how to prove this....
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8 views

Inverse Fourier analysis problem - matlab

Using MATLAB, I am trying to do an analysis of a triangular pulse and recreating it from acquired coefficients of the Fourier series. The pulse looks like this ...
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1answer
40 views

Dirichlet's Theorem for Turan's Method of Analysis

This question concerns a special case of Turan's power sum, which looks reasonably straightforward. Let $S(N,\nu)=\sum_{n=1}^N z_n^\nu,$ where all the $z_n$ are on the complex unit circle. Then, ...
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1answer
60 views

Hilbert Transform: limit of xHf(x)

In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that $\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$ where $f$ is a Schwartz function and $H$ is the Hilbert ...
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Fourier analysis. How to present a numerical solution of a nonlinear differential equation in the amplitude-frequency spectrum?

I have a system of nonlinear differential equations: ...
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12 views

Prove a relationship established between the norm of a norm of a function and that of its derivative

From the book Fourier Analysis an Introduction Chapter 3 Exercise 11 b,c b) If $f$ is $T$-periodic, continuous, and piecewise $C^{1}$ with $\int_{0}^{T}f(t)dt=0,$ and $g$ is just $C^{1}$ and $T$-...
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2answers
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Decomposing Sums of Random Variables

Suppose I have $M$ random variables, and a number of realizations of each variable. Each RV has the probability mass function: $$\rho_{X_i}(x) = \begin{cases} p_i, & x = 1\\ 1-p_i, & x = 0\\ 0,...
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Relation between Radon transform and Fourier transform

I am reading the notes provided here: https://link.springer.com/content/pdf/10.1007%2F978-1-4419-6055-9.pdf which I have some questions about. On page 2 of chapter 1, the Radon transform of a ...
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2answers
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Measurability of maximal functions

I have been reading Fourier Analysis by J. Duoandikoetxea. I got stuck on proving the measurability of maximal functions. The general maximal function/operator in this book is from the following ...
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Equality between sum and convolution

Let be the sum $$A(r,n)=\sum_{k=0}^{n-1}k^r(n-k)^k$$ Definition of discrete convolution of function $f$ to itselfs in point $n$ is $$(f*f)[n]=\sum_{m=-\infty}^{\infty}f[m]f[n-m].$$ Looking to the $A$ ...
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1answer
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Detecting Uniform Convergence By Fourier Methods

A close snorbaki friend has posed a problem about uniform convergence of a sequence of functions, and I've noted it has a particularly well behaved sequence of Fourier transforms. Can I dazzle my ...
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1answer
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Fourier series of $f(x)=\int\limits_0^x \ln\sqrt{\frac{1}{2}\left| 1+\sqrt3 \tan\frac{t}{2} \right|} \ \text dt$

Find the Fourier series of the function $$f(x)=\int\limits_0^x \ln\sqrt{\frac{1}{2}\left| 1+\sqrt3 \tan\frac{t}{2} \right|} \ \text dt$$ or show that it does not exist. The first thing I have ...
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0answers
28 views

Parseval's theorem for the Hankel transform

Consider, for a fixed $n \geq 0$, the "Hankel-type" transform $$\hat{f}(k)=(2\pi)^{-(n+2)/2}\int_0^\infty dx\,x^{n+1}\left(\int_0^\pi d \theta\,\sin^n(\theta) e^{ikx\cos(\theta)}\right)f(x)$$ a priori ...
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0answers
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Some explanatios in Wollf's Harmonic Analysis notes

I read these notes in harmonic analysis: http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf I started to study the proof of proposition 8.2 (page 54) The author constructs a measure using ...
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3answers
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How to find functional square root of $\sin(x)$.

Maybe I am overlooking something, but is there some easy way to find a function $x\to f(x)$ so that $$(f\circ f)(x) = f(f(x)) = \sin(x)$$ on some interval, say $x\in [-\pi,\pi]\subset \mathbb R$ of ...
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calculate the Fourier transform of $\frac{1}{(x-k)^4}$, $k$ not real

Consider $\int \frac{1}{(x-k)^4}e^{-ixy}dx$, and assume that $k$ lies in the upper half plane. Then consider the semicircle in the upper half plane with its straight side lying on the real line. The ...
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Fourier transform of frequency division [migrated]

I need a mathematical description of what happens to the spectrum of a signal when that signal is processed to divide its frequency. As an example, construct an input signal as the sum of two sine ...
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1answer
36 views

Does any $L_1(-\pi, \pi)$ function whose Fourier coefficients are 0 equal 0 a.e.?

Suppose $f \in L_1(-\pi,\pi)$, satisfying $\hat{f}(n) = 0, \forall n\in \mathbb{Z}$ (which means all Fourier coefficients are $0$). Does $f = 0$ almost everywhere in $(-\pi,\pi)$? The Fourier ...
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2answers
29 views

Fourier Series Exponential Representation

As part of my education, I took upon myself to understand where the Fourier series functions come from, I did some digging, and found out that the vector space accommodating the function is a Hilbert ...
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2answers
66 views

Show that a sequence is not Cauchy in $C^\infty_c(\mathbb{R})$

Denote $C^\infty_c(\mathbb{R})$ the space of all compactly supported infinitely differentiable functions. We equipped $C^\infty_c(\mathbb{R})$ the topology induced by the following family of semi-...
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0answers
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Estimate for the Fourier coefficients

In the book I am reading, the author gives an estimate for the Fourier coefficients and I don't get it. In the following we consider integrable functions $f \in L^1(S^1)$ which are measurable ...
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0answers
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Understanding a proof : Fourier transform in distribution sense.

Let $u\in L^1(\mathbb T^n)$ on the torus. The Fourier transform of $u\in\mathcal S'(\mathbb R^n)$ (the temperated distribution) is given by $$\hat u=\sum_{k\in\mathbb N}c_k\delta _k,$$ where $c_k$ are ...
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1answer
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Rudin's Construction of Inductive Limit Topology: unnecessarily abstruse?

In Rudin's Functional Analysis Book, one of the examples in the first chapter is used later in the chapter on distributions. But when he gets to defining the inductive limit topology on a certain ...
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What are Fourier Coefficients with respect to a circle action

I am aware of the typical Fourier coefficients of a $2 \pi$ periodic function $f$ given by it's $L^2$ inner product with the basis $e^{i n \theta} $ $n \in \mathbb{Z}$. However, an article I'm reading ...
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2answers
54 views

Use symmetry and odd/even function considerations to find the Fourier series of $f(x)=\cos\left(\frac{x}{2}\right)$ on $[-\pi,\pi]$

Show that the Fourier series of $f(x)=\cos\left(\frac{x}{2}\right)$ for $-\pi\lt x \lt \pi$ is given by $$f(x)=\frac{2}{\pi}+\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\cos(nx)}{4n^2 - 1}$$ ...
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1answer
23 views

Matrix Representions [closed]

Let $ V=\{ f\in\text{func} (\mathbb{R}, \mathbb{C}) : f(t) =\alpha \cos (t) +\beta \sin(t), \alpha, \beta \in \mathbb{C} \} $. (a) show that cos$(t) $, sin$(t) $, and exp(-it), exp(it) both ...