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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Sequence sum + convolution .

Definition: Let be $u$ and $v$ two sequences the convolution of these sequences is defined than $$h(m)=u(m)* v(m) = \sum_{s=-\infty}^{\infty}u(m-s)v(s).$$ Question: Show that ...
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Fourier Transformation

This expression: $x(t)=[e^{-3t+5}] u(t-1)$. I am trying to take the Fourier transformation of the above expression. I know that for $x(t)=[e^{-at}] u(t) \leftrightarrow \frac1{i\omega+a}$. But, ...
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80 views

Fourier analysis with Galois Theory

Do you know some combinations between Galois theory and Fourier analysis and what are the applications?
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85 views

DFT of basis functions

Suppose functions $u_{1}(x)$,..$u_{K}(x)$ are a basis of $H[0,1]$( some space of real-valued functions). Define Discrete Fourier transform $$ U_{l}(x)=\sum_{j=1}^{K}u_{j}(x)\exp(2\pi i lj/K) $$ and ...
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Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$

I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into ...
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Why use Hilbert transform for non-stationary time-series

Why is the Hilbert transform preferred over the Fourier transform for non-stationary time series (like amplitude modulated radio signal)?
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252 views

operator norm of this multiplier operator

I am having some trouble with some basic properties of a given operator. Firstly, the operator T is defined as taking the fourier inverse transform of the function ...
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561 views

Inverse Fourier transform of a hyperbolic cosine

This problem arises from trying to solve, by Fourier transform, the Cauchy problem $$\begin{cases} u_{tt}-u_{xxxx}=0 &x\in\mathbb{R},\, t\geq 0\\ \begin{cases} u(0,x)=f(x)\\ u_t(0,x)=0 ...
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242 views

Negative exponent Fourier Transform of Sequences

Why the exponent must be a negative in the Fourier transform of any sequence? What happens with expressions $$x(m)=\dfrac{1}{2\pi}\int_{-\pi}^{\pi}X(w)\exp(jmw)dw$$ if we define the Fourier ...
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247 views

How to find the coefficient of this Fourier sine series?

From $$1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$$ I want to find $a_k.$ My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, ...
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143 views

Find the Hardy-Littlewood maximal function of $\chi_{[-1,1]}$ on $\Bbb R$

Find the Hardy-Littlewood maximal function $Mf$ of the $\Bbb R^\Bbb R$ function $f=\chi_{[-1,1]}$. How do we find $Mf(x)$ for $|x| > 1$? I see that it should decrease like $1/x$, but I can't find ...
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821 views

Fourier Sine Transform of $e^{-ax^2}/x$

I'm trying to do this integral essentially: $$\int^\infty_0 \frac{e^{-ax^2}\sin(kx)}{x}dx$$ which I realized to be $$\frac{1}{2}\operatorname{Re}\left[F\left(\frac{e^{-ax^2}}{x}\right)\right]$$ where ...
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106 views

Conjugate pairs in Fourier transforms but with Fourier coefficients

If $f(x)$ and $g(\alpha)$ is a pair of Fourier transforms, then how can we show that $df/dx$ and $i\alpha g(\alpha)$ is a pair of Fourier transforms?
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511 views

Give an example a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge

Give an example of a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge. That is, for any $a,b \in \Bbb Z_+$, $$ \|f_n\|_{a,b} < \infty, $$ but $$ ...
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85 views

Integral of Scaled Bessel Function With Linear Phase

I am trying to solve a problem part of which includes the following integral ($j=\sqrt{-1}$): $$\int_{k_1}^{k_2} k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k$$ The $e^{-jk\sigma}$ term is making my ...
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Show the Fourier transform is continuous in the Schwartz space $\mathcal S(\Bbb R)$

Show the Fourier transform $\mathcal F$ is continuous in the Schwartz space $\mathcal S(\Bbb R)$. Use the standard $\mathcal S$-norms $$ \|f\|_{a,b}=\sup_{x \in \Bbb R} \left| x^af^{(b)}(x)\right|, \, ...
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87 views

Theorem Fourier Analysis

The inner product of the two-dimensional sequences $f(x,y)$ and $g(x,y)$ is equal to the inner product of their Fourier transforms, that is: ...
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118 views

Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
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111 views

The Continuity of the Discrete Time Fourier Transform of Absolutely Summable Series

I saw on a book to following claim: Given an Absolutely Summable Series $ \sum_{n = -\infty }^{\infty}\left | x\left [ n \right ] \right | \leqslant \infty $, Namely, $ l_1 $ series it is possible to ...
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247 views

Show a Schwartz function vanish at infinity

Let $f$ be in the Shwartz space $\mathcal S(\Bbb R)$. Why does the $\mathcal S$-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)|, \text{ for } a,b \in \Bbb Z_+, $$ implies that $f$ vanish at ...
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bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
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70 views

A question about Fourier transform

I just don't know how to calculate the the fourier transform of $1/(1+x^2)$.Can you help me guys? Thx
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858 views

Show smooth functions of compact support are dense in the Schwartz space

Show $\mathcal{D}=C_c^\infty(\mathbb R^n)$ is dense in the Schwartz space $\mathcal{S}(\mathbb R^n)$. Use the standard topology on $\mathcal{S}$ $$ \|f\|_{a,b}=\sup_{x \in \Bbb{R}^n}\left| ...
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270 views

Why doesn't repeating a signal give rise to a finer resolution of DFT/FFT?

If x = [1 2 3 4 3 2]; and x1= [x x x x x x x x x]--that a new vector made of duplicating copies of x, then why is it that the FFT of x and x1 are essentially the same. When I plot the FFTs of each ...
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251 views

Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$

Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $$ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $$ But ...
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533 views

Show the usual Schwartz semi-norm is a norm on the Schwartz space

Let $f \in C^\infty(\mathbb R)$. Define the semi-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)| $$ where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$. Show ...
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159 views

Question about Fourier Transform

I am reading a Fourier Transform definition in two places, in the first is $$\int_{-\infty}^{\infty}f(x)\exp(-ijw)dx$$ and another is $$\int_{-\infty}^{\infty}f(x)\exp(-2\pi ijw)dx$$ I want know ...
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149 views

Question on Parseval's theorem

If $\sum_{k=-\infty}^{\infty}|a_k|^2$ is not finite, does Parseval's theorem say that the Fourier transform of $a_k$ is also not finite?
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Regularizing effect of the heat equation

Consider the heat equation on $\mathbb{R}_+\times\mathbb{R}^d$ \begin{align*} \partial_t u -\Delta_x u &= f, \\ u(0,x)&=u_0(x). \end{align*} In the case where $u_0\in L^2(\mathbb{R}^d)$ ...
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Suggestion for a project on Harmonic measure and Fourier analysis

I have a course project on harmonic measure and Fourier analysis. The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis. Harmonic measure is a vast ...
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557 views

Inverse fourier transform of $f(x)=\frac{\text{u}(x)}{\sqrt{1-x^2}}$

How to calculate (or derive) the inverse Fourier transform of $$f(x)=\frac{\text{u}(x)}{\sqrt{1-x^2}}$$ where $u(x)$ is the rectangular function? I know that $f(x)$ is the Fourier transform of the ...
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129 views

Violation of Parseval's theorem?

Can a function $f:G\to\mathbb{C}$ in $L^p,\ p>1, p\neq 2$ have a Fourier transform $F:\hat{G}\to\mathbb{C}$, where $\hat{G}$ is the Pontryagin dual space of $G$? I believe it can be shown that such ...
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Schwartz space: semi norm estimate on translation

the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$: $$ \|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| ...
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226 views

integral evaluation of an exponential

let be the function $$ e^{-a|x|^{b}} $$ with $ a,b $ positive numbers bigger than zero then how could i evaluate this 2 integrals ? $$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$ here 'c' can ...
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101 views

Proof for Fourier transform in $L^2$

This question makes me really confused: Let $f$ and $g$ two functions in $L^2$. Show that: $$\int \widehat f\cdot gdx= \int f\cdot\widehat gdx,$$ where $\widehat f$ is the Fourier transform ...
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119 views

$L^p$ space and Fourier Transform

Let be $q>p$ then $L^q(\Omega)\subset L^p(\Omega)$. I will be able to say that all $f \in L^q$, such that $q>1$, have a Fourier Transform?. pdta:I asking this because I am read that exist ...
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433 views

Discrete Fourier transform of shifted N-periodic sequence

I'm trying to show that if f[n] is an N-periodic sequence then the Discrete Fourier Transform of the shifted sequence f[n-m] for some constant $m\in\mathbb{Z}$ is $e^{\frac{-2\pi i mk}{N}} F[k]$ ...
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So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?

For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation} H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\}, \end{equation} where $\mathcal{S}'$ is the space of ...
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129 views

Fourier Series and Filter in Function

Let $L=1, A=1$ and $f\in L^2([-L/2, L/2])$, with Fourier series $$f^{t}=\sum_{n=-K}^{K}a_n \exp(2j\pi xn/L),$$ truncated at $K$. Has this function, $f^{t}$, any relation with Fourier Inverse ...
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Dual space of the function $f$ in Fourier Transform

Let $f\in L^1{(\mathbb{R})}$. Why the Fourier Transform $\hat{f}\in L^{\infty}{(\mathbb{R})}$. Is it because $(L^1{(\mathbb{R})})^*=L^{\infty}{(\mathbb{R})}$?
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Proving/disproving an identity on a Hessian.

Let $A=(a_{i,j})$ be a $n$ x $n$ matrix $(n\geq 2)$, where $a_{i,i} = |x|^2-2x_i ^2$ and $a_{i,j} = -2x_i x_j$ for $i\neq j$. Here $|x|^2 = x_1^2+x_2^2+ \cdots + x_n^2$. I'd like to compute the ...
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207 views

Expression in Fourier Transform

Let be $f\in L^1(\mathbb{R})$, I will be able to say that $$ \dfrac{\hat{df(w)}}{dx} = \int_{-\infty}^{\infty}\dfrac{df(x)}{dx}\exp(-2\pi j wx)dx $$? Why?
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Taking inverse Fourier transform of complicated multipart equation

Define $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} ...
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Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
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Expressing the cantor function on $[0,1]$ as a function on $\text{Ternary}([0,1])$

I would to link the simple function and probabilistic approach for the calculation of the Fourier transform of the Cantor function. Let $f:[0,1] \to [0,1]$ be the Cantor function. In the simple ...
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Heat equation, separation of variables and Fourier transform

I have a question about the heat equation $\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}$ with the conditions that $\varphi(x,t=0) = f_0(x)$ and $\lim_{x \rightarrow\pm ...
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Difference between Fourier series and Fourier transformation

Whats the difference between Fourier transformations and Fourier Series? As I've been working with Fourier Series in my maths lectures yet a friend of mine also doing engineering has been working with ...
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263 views

Fourier transform of a Generalized Gaussian

I've got a family of functions called Generalized Gaussians. They're given by: $f(x) = \exp(-ax^{2p})$ Where $p \in \{1,2,3,\ldots\}$ Could anyone tell me how to find their Fourier transforms?
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85 views

Probabilistic calulation of the Fourier transform of the Cantor function

This is on the same theme as in this post, where the Fourier transform was derived using simple function. Let $f:[0,1] \to [0,1]$ be the Cantor function. Then $f$ is the cumulative distribution of ...
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1answer
429 views

Deriving complex form of Fourier series

I have encountered a problem where I get the correct outcome, but I am uncertain as to whether or not my steps are logically justified. I would really appreciate some input regarding this! The ...