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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Is it true that the Fourier coefficient of convolution is the product of the coefficients?

what I mean by the title is the following: if we define the convolution between two $2\pi$-periodic, $C^1$ functions as $f*g(x) = (2\pi)^{-1}\int_{-\pi}^\pi f(x-y)g(y)dy$, is it true that the Fourier ...
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35 views

Questions about $\delta$ Dirac function-Fourier transform

In my notes there is the following: $$f(x)=\delta(x-x_0)$$ $$\text{The more closed a function is, the spectrum will be wider.}$$ $$\widetilde{f}(k)=\int_{-\infty}^{+\infty}{\delta(x-x_0)e^{-ikx}}dx$$ ...
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13 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
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18 views

about lp estimate of schwarz function

A homework question that I really couldnt find how to start. Prove that for any f in schwarz class $ \lVert f \rVert_{q} \leq C_{p,q} \lVert \nabla f \rVert_{2}^{a} \lVert f \rVert_{2}^{1-a} $ $ ...
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fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
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10 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetical functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
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When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
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21 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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15 views

Fejer's Theorem in relation to the Fourier Transform

I have this question that relates the Fejer theorem with the Fourier Transform. Any help would be appreciated. If $f$ is of moderate decrease then $$\int_{-R}^{R}\left(1-\frac{|\xi|}{R}\right) ...
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What is the discrete Fourier transform of the sets $(1,0,0,0)$ and $(0,1,0,0)$?

What is the discrete Fourier transform of the sets $(1,0,0,0)$ and $(0,1,0,0)$? I am unable to understand the progression from a continuous Fourier transform to a discrete Fourier transform. ...
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16 views

WHY?The Squared Euclidean Imbalance are equal to Fourier coefficients

I'm reading the classical paper about distinguishing attack, How Far Can We Go Beyond Linear Cryptanalysis ,Thomas Baign`eres, Pascal Junod, and Serge Vaudenay. The only proposition I don't ...
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33 views

Is the Fourier transform of the function correct?

I am asked to calculate the Fourier transform of the function $$e^{-\frac{x^2}{2a^2}}, a>0$$ $$$$ After calculations I have found the following: $$a \sqrt{2 \pi} e^{-\frac{a^2 k^2}{2}}$$ $$$$ Is ...
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21 views

Need help to understand calculating fourier transform of 1

this issue is related to a physics problem, but since it is mathematical I will post it here. When calculating the following Fourier transform $$ -i\int_\infty^\infty dt~ e^{i\omega ...
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19 views

recover sin from phase and current amplitude

I'm having hard time with this as math is not my everyday job. So I have fft from two consecutive samples. I have chosen one frequency from the spectrum and I need to recover a sinus that generated ...
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23 views

discrete fourier transform proof (show equals n*I)

Let $w=e^{(-2\pi i/n)}$. Let $W$ be an $n \times n$ matrix defined by $$ W = \begin{pmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & w & w^2 & w^3 & \cdots & ...
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13 views

Find the function $f(x)$ when it satisfies the ode

The Fourier transform of the function $\frac{d}{dx}f(x)+xf(x)$ is $i[\frac{d}{dk}\widetilde{f}(k)+k\widetilde{f}(k)]$, so if a function $f(x)$ satisfies the ode $\frac{d}{dx}f(x)+xf(x)=0$, then the ...
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13 views

Obtain the complex Fourier Series of the following function:

$$f(t)=t^3 \;\;\;\;\;\;\;\;\;\;\;\; -3/2<t\leq 3/2 $$ $$f(t)=f(t+3)$$ I've tried setting up an integral for $C_n$ coefficients using the formula $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ...
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12 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
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30 views

Alternative proof of Heisenberg uncertainty principle - step 1

I'm student of physics having problem with pdes. I have rapidly decreasing function in $\mathbb{R}^d$ st $\int{|u|^2}dx=1$ and function $v=e^{i\langle\psi\rangle x}u(x+\langle x\rangle)$, where ...
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30 views

Evaluate the following Dirac delta integrals:

a) $ \int^{+\infty}_{-\infty} \delta'(t-\pi)e^{-t^2} \; dt$ b) $ \int^{+\infty}_{-\infty} \delta(-3t)(\frac{e^{-t^2}}{\ln(t^2 + 3)}) \; dt $ c) $ \int^{+\infty}_{-\infty} \delta(4t)\sinh{t^2} ...
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Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
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24 views

Calculate the Fourier Transform of the function

I have to calculate the Fourier transform of the function $f(x)=sgn(x) e^{-a |x|}, a \geq 0$. After that I have to take the limit $ a \rightarrow 0$ to calculate the Fourier transform of the sign ...
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42 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
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16 views

Density of a set of functions in Schwartz space

I have a difficulty doing the following problem: Let $S(\mathbb{R}^n)$ be the Schwartz space. I need to determine whether the following set of functions $A$: $$A= \{f\in S(\mathbb{R}^n): ...
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1answer
37 views

How can I prove this property?

I need some help... I am asked to prove the following property of the Fourier transform, when $F[f(x)]=\widetilde{f}(x)$, where $F[f(x)]$ is the Fourier transform of $f(x)$: $$F[ \widetilde{f}(x) ...
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1answer
25 views

The derivative of a function is square integrable assuming Fourier transform dominated

I am struggling in solving the second part of this problem. Let $g$ be a continuous function in $L^1(\mathbb{R})$ whose Fourier transform is the function $F$. Suppose $|F(x)|\leq (1+x^2)^{-2}$. Prove ...
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24 views

Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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9 views

Appropriate frequency domain transform for bounded periodic complex waves

I'm working on a wave inversion problem for which the data take the form of 2D spatial frequencies within boundaries. My understanding is that the appropriate frequency-domain transform for a periodic ...
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33 views

Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
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solving a complex fourier series

while solving for co efficient of a complex fourier series i came across this form cos(0.5*n*pi)+i*sin(0.5*n*pi) is there any way to simplify this? Note: N is an integer
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1answer
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Characteristic Function as Fourier Multiplier

In lecture notes I have, it is mentioned that the characteristic function $\chi_I$ of an interval in $\mathbb{R}$ is an $L^p$ Fourier multiplier for $1 < p < \infty$. I thought this would be ...
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1answer
34 views

Integral equation, Fourier transform

Find all functions $ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve $\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$, $ t\in \mathbb{R}$ How do I solve this? I know that the left part is the ...
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18 views

fourier-transform: where is my mistake?

I am trying to do a fourier-transform of the function $$\psi(x,t=0) = \frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}e^{-\frac{x^2}{4\sigma^2}}e^{ik_0x}$$ My calculation is $$\int_{-\infty}^\infty ...
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1answer
20 views

Fourier series with a weighted mean square norm

I am interested in Fourier series with a non-uniformly weighted error norm. What I mean by this is that the usual Fourier series of a periodic function is a minimizer of the mean squared error: $$ J_N ...
2
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38 views

(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
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14 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
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28 views

Basic question about the Discerete Fourier Transfrom

I have trouble understanding the transition from the infinite integral of the Fourier transform $$ \mathcal{F}f(v) = \int^\infty_{-\infty}e^{ivk}f(k)dk $$ to the discrete version $$ \mathbf{F}f_n = ...
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1answer
19 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
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29 views

Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
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1answer
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Please help with this Discrete fourier transform question

Consider the ODE $\frac {d^2u}{dx^2} + 2\pi\frac {du}{dx} + \frac 54\pi^2u = g(x)$ where g is a periodic fuction with period 1 given by $g(x) = e^{\pi x}$ , $ 0 \le x \lt 1$. It is desired to find ...
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24 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
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26 views

Triangular signal Fourier transform. Please help me [closed]

Please help me. I don't know how to solve part b) (the triangular signal fourier transform).
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26 views

What can be observed by evaluating a polynomial at roots of order greater than the polynomial itself?

I have been reading through an algorithms book on the use of FFT for large number multiplication. An example it used to emphasize a point was: Evaluate the following polynomial at all roots of unity ...
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27 views

Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
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1answer
30 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
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1answer
14 views

how to prove translation identity: $(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x)$?

Let $m, f\in S(\mathbb R^{n})$=The Schwartz space. My question: How to prove: $$(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x);$$ where ...
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20 views

properties of frequency- decomostion operator $\square_{k}^{\sigma}=\sum_{|\ell|_{\infty}\leq 1}\square_{k}^{\sigma}\square_{k+\ell}^{\phi}$?

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
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1answer
18 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
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1answer
25 views

testing out the formula. I don't know how to intepret this!

$$\frac{d^{2}u}{dt^{2}}-4\frac{d^{3}u}{dt dx^{2}}+3\frac{d^{4}u}{dx^{4}}=0$$ $$u(x,0) = f(x)$$ $$\frac{du}{dt}(x,0) = g(x)$$ Relevant equations The attempt at a solution First I use fourier ...
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8 views

Nyquist Sampling Theorem

I'm working on the proof of the Nyquist sampling theorem. Mainly I'm wondering about the regularity conditions. In particular, suppose $f$ is continuous on $\mathbb{R}$ and $\hat{f}(k)=0$ unless ...