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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Solving a simple trigonometric equation for coefficients

Is it possible to solve the equation $$ a \sin x + b \cos x + c \cos^3 x = d \cos x $$ where $c\neq0$ using some coefficients $a$, $b$, $c$, and $d$? I can't see how to make the frequency of ...
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35 views

Derivation of Fourier transform of Heaviside function

I read some proofs of Fourier transform of Heaviside function on this site but I don't really understand (because I haven't learned about distribution yet). I'm trying to derive it myself based on ...
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42 views

evaluation of fourier transform of electric potential

I would like to ask how to evaluate equation 7? I have spent hours and still have no idea how to get a(k).
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29 views

Parition of unity argument in a Fourier analysis paper

I am currently reading a paper "The proof of the $\ell^2$ decoupling conjecture" by Bourgain and Demeter. I was wondering if someone could explain me one of the arguments in their paper. First I will ...
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11 views

Fourier Computation integral

$$F(w)=\int_{-\infty}^{\infty} e^{-|x|+ix}e^{-iwx} \, dx$$ Need help to compute.
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$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : ...
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31 views

Fourier Transform of a shifted & scaled rectangular pulse

I'm trying to find the Fourier Transform of the following rectangular pulse: $$ x(t) = rect(t - 1/2) $$ This is simply a rectangular pulse stretching from 0 to 1 with an amplitude of 1. It is 0 ...
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43 views

Algebraic way to see why only $n=3$ is a valid coefficient

I'm a bit of a sucker for brute force calculations. Say I want to calculate a coefficient with Fourier theory, in my case \begin{align*} a_n = \int_0^1 \sin (3\pi x) \cos (n\pi x) dx. \end{align*} ...
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32 views

Particular solution to a nonhomogenous differential equation

Consider the following ODE: $$\frac{d^2 x}{dt^2} + kx = f(t)$$ where $$f(t) = \frac{1}{2} +\sum_{n = 1}^{\infty}{ \frac{-4}{n \pi} \sin\left(\frac{n \pi t}{2}\right)}$$ was derived using fourier ...
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Inverse Fourier Transform of $\left| \cos\left(\frac{2 \pi f}{100}\right) \right|$

I made 2 approaches: Am I any close...how do i procced?
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19 views

Fourier transform , of a function computation

I need help with this Fourier transform computation. $$F(w)=\int_{-\infty}^{\infty} e^{-|x|+ix}e^{-iwx} \, dx$$ Need help to compute.
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48 views

Inverse Fourier Transform of | cos[(2 pi f)/100)] |

I would like a help calculating the Inverse Fourier Transform of Absolute cos[(2 pi f)/100]
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Fourier transform of phase shifted cosine function

Fourier transform of function $\cos(w_m-w_0)t$ is $\pi [ \delta (w-(w_m-w_0)) ] + \delta (w+(w_m-w_0)) $. What would be FT of $\cos((w_m-w_0)t+\phi)$?
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30 views

Fourier transform

I am having a difficulty understanding how to approach this problem. I have state vector $$f(x)=e^{-|x|+ix}$$ and observable $$P=-id\dfrac dx $$ I want to find fourier transform of $f$? and the ...
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36 views

Fourier transforms having compact support

As we know, the fourier transform is a map $\mathcal{F}:L^1\rightarrow C_0$ (all with domain $\mathbb{R}$). Can one characterize the space of $f\in L^1$ such that $\mathcal{F}$ has compact support, ...
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46 views

Fourier transform and convolution

Let $f \in L^1(\textbf{R})$ be such that $f'$ is continuous and $f' \in L^1(\textbf{R})$ . Find a function $g \in L^1(\textbf{R})$ such that $$ g(t) = \int_{-\infty}^{t}e^{u-t}g(u)\,du + f'(t) $$ ...
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113 views

Fourier Transform of a line $Ax+By+C = 0$

Can someone help me in a step-by-step derivation for the Fourier Transform of a line ? It appears to be simple but still cannot figure out. I know what is the end result but I am unable to figure out ...
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2D linear schrodinger equation

I am trying to solve a 2D Schrodinger equation of the following form. This is in the context of Partial Differential Equations. \begin{align} iu_t + \frac{1}{2} (u_{xx}+u_{yy}) & = 0 \\ ...
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How to get the equality on potential kernel?

Recently, I am reading the book ``Some random series of functions" by Jean-Pierre Kahane. I cann't understand an equality on page 134, Chapter 10. Namely, let $\mu$ be a probabiliy measure with ...
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39 views

What is a “standard symbol” in analysis?

I was reading some Wikipedia pages about analysis when I came across this strange "standard symbol" terminology in the "Fourier integral operator" page. It seems to be a function (or a distribution) ...
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32 views

Convolution and Fourier transform

Im stuck at a rather simple question. The problem is this Solve the integral $$ \int_{-\infty}^{\infty} \frac{\sin [5(t-u)]\sin 6t} {u (t-u)}du $$ And this is just the convolution of $$\frac{\sin ...
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20 views

Application of time reversal property of DFT

I came across the time reversal property of $DFT$ that states: \begin{equation*} x(<-n>_{N})\rightarrow X^{*}(k) \end{equation*} I can't seem to understand though how can this property be used ...
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17 views

Is Fourier transform has support in given interval?

Given an finite interval $(a,b)$, what is the guarantee for existence of a $L_2$ function on $\mathbb{R}$ such that whose $L_2$ Fourier transform has support in $(a,b)$?
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31 views

Why does knowing whether the function is odd or even make calculating the fourier series easier? [duplicate]

The fourier series is given by $$f(x)=\frac12a_0+\sum_{n=1}^\infty(a_n\cos(nx)+b_n\sin(nx))$$ after doing a few examples I noticed that whenever I was evaluating an even function the $a_n$ would ...
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37 views

Fourier transform with second order differentials

How do I start this question: $$\frac{d^2y}{dx^2} = yx$$ for a function $y(x)$ which tends to zero as $x \to \pm \infty$. Show that transform of $\hat{y}(k)$ of $y(x)$ satisfies the first order ...
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Can the system of shifts of an $L^2(\mathbb{R})$ function be an ONB?

In Wavelet theory, one constructs wavelet bases via translations a dialations of an $L^2$ function... Is it possible for some set of translations alone to form an Orthonormal Basis? That is: Does ...
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34 views

Pointwise convergenve of mollified $f\in L^1_{loc}$

Let $\Omega\subseteq\mathbb{R}^n$ open, $f\in L^1_{loc}(\Omega)$, $\eta_\epsilon(x) = \dfrac{1}{\epsilon^n}\eta(\dfrac{x}{\epsilon})$ the usual scaled mollifier, i.e. $supp (\eta_\epsilon) ...
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67 views

Fourier transform of the identity function $f(x)=x$

Let's say you are given $\omega_f \in \mathcal{S}'(\mathbb R)$ with \begin{align*} f \colon \mathbb{R} &\to \mathbb{R}\\ x &\mapsto x, \end{align*} and the definition of Fourier transform ...
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27 views

Is a bandlimited function smooth?

Definition of smoothness Definition of bandlimited: for simplicity, let's consider a function in $L^2(\mathbb{R})$ is bandlimited if the support of its Fourier transform lies in $[\Omega, \Omega]$ ...
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Phase of the Fourier Transform of a function

The Fourier Transform of this function \begin{equation} f(n)=u(n)-u(n-m) \end{equation} (where $u$ is the unity step) is: \begin{equation} F(\omega)=\frac{\sin(\omega ...
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Fourier transform of dirac by two methods

I would like to apply the fourier transform of the differential and I chose the following example:The heaviside function is $h(t)=1$ if $t\ge 0$ and $h(t)=0$ elsewhere. The differential of $h$ is ...
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28 views

Uncertainty principle density argument

I proved the Heisenberg Uncertainty Principle for $f$ in the Schwartz space $ S(\mathbf R)$: $$ \int_{\mathbf R} |\xi \hat{f}(\xi)|^2 \int_{\mathbf R} |xf(x)|^2 dx \geq \frac{1}{(4\pi)^2} ...
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298 views

Intuition behind the DTFT vs Fourier transform of ideally sampled signal

So I am taking a signal processing course in EE and my professor is an Engineer who reallly likes math however his book which we use for the class falls in the dreadfull purgatory of math books in my ...
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36 views

Fourier Transform and Trigonometry polynomials

Let $f$ be a continuous function on $\mathbb{R}$ with period 1. Let $\theta \in \mathbb{Q}^c$, then $$\lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n=1}^N f(n\theta) = \int_0^1 f(t) dt.$$ I think that ...
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Matrix form of Fourier and Fresnel Transform

I am wondering how to write up the matrix form of, say, Fresnel or Fourier transform. I know that for the case of Fresnel it would be Toeplitz matrix.
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What is the advantage of the Fourier Transform over the Hartley Transform?

The Hartley_transform is defined as $$ H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t) \, \mbox{cas}(\omega t) \mathrm{d}t, $$ with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega ...
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Law of large numbers for continuous periodic functions

Suppose $f$ is a continuous function on $\mathbb{R},$ with period 1. Prove that $$ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f\left(n \theta\right)=\int_{0}^1 f(t) \ dt $$ for every ...
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Integral of Fourier exponential over any measurable set converges to zero

I am trying to solve the following problem. Show that if $A \subset \left(0,1\right)$ is measurable, then $\int_{A} \exp \left(2\pi ikt\right) \ dt \xrightarrow{k \rightarrow \infty} 0$. My attempt: ...
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18 views

Solve this integral equation using Fourier transform

Solve this integral equation using Fourier transform $$\int_{-\infty}^{\infty} \frac{f(t)}{(x-t^2)+a^2} dt= \frac{\sqrt{2} \pi}{x^2 + b^2}$$ for $b> a > 0 $ Please Help see my answer ...
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Fourier transform of $\frac{1}{a+b\cos(d x)}$

Is it analytically possible to calculate the Fourier transform of $$\frac{1}{a+b\cos(d x)}$$ where $a, b$ and $d$ are constants and $x$ is the variable. If so, how? Or at least what's the approach? ...
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1answer
40 views

Critical Homogeneous Sobolev Embedding

For $s\in\mathbb{R}$, $1<p<\infty$, define the homogeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^{n})$ as follows: For $f\in\mathcal{S}_{0}(\mathbb{R}^{n})$ (Schwartz functions with Fourier ...
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17 views

Solving linear dispersive wave equation with Fourier transform

I want to solve the following linear dispersive wave equation. $$u_t+u_x+u_{xxx}=0 \tag{1}$$ In particular, I want to see how a simple waveform evolves over time. Because it's linear, I know the ...
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33 views

Solving PDE with Separation of Variables

I have done a few of these now but I'm stuck at this one $$\begin{array}{} u_{xx}=u_{tt}+2u_t \\ u(0,t)= u(\pi,t)= 0 \\ u(x,0)=0, \, u_t(x,0)=\sin ^3x \\ \end{array} $$ for $0<x< \pi$ and ...
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Solutions of an EDO in tempered distribution space being smooth out of the origin.

For $a\in\mathbb{C}$, let us consider the following differential equation in $\mathcal{S}'(\mathbb{R})$, the set of tempered distributions on $\mathbb{R}$: $$xT''+2T'+(a-x)T=0.$$ For ...
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Hyperbolic or exponential solutions to differential equation

I have spent the last couple weeks in my Fourier Analysis course to solve PDEs with the method of separations of variables. However, I have come up with something that annoys me and I can't really ...
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25 views

Magnitude and Angle of Discrete Fourier Transform

I can't figure out how to get the magnitudes for periodic discrete Fourier transforms. For example if $x[n] = cos(\frac{\pi}{4}n + \frac{\pi}{2})$, I need to find and plot the magnitude ...
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1answer
30 views

Fourier Transform of the “regular” tempered distribution of $|x|$

As the title states I am trying (without luck) to compute the Fourier Transform (in tempered distributional sense) of $|x|$, meaning ($\mathcal{S}(\mathbb{R})$ the Schwartz space): ...
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Estimating number of terms for partial fourier sum to minimize error

A function is define as $$ f(t) = \begin{cases} 0 & \text{ if } \pi<x\le -1 \\ x^2 & \text{ if } -1<x<1 \\ 0 & \text{ if } 1\le x < \pi \end{cases} $$ Find the ...
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Getting sum of $2 \pi$ periodic function

I have a $2\pi$ periodic function which in the interval $[0,\pi]$ is $f(t) = \sin{\frac{t}{2}}$. I have to find the sum for $t \in \mathbb{R}$. But do I know anything about $f(t)$ outside of $t \in ...
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Example for Fourier transform in $C_b^0$ [duplicate]

$C_b^0(\mathbb{R})=\{f\in C(\mathbb{R}),\lim\limits_{|x|\to\infty}f(x)=0\}$ So can we construct an function in $C_b^0(\mathbb{R})$ such that it's Fourier transform is not in $L^1(\mathbb{R})$? What ...