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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58 views

A question on when Fourier Transform on the circle get vanished.

I am given $f\in L^1 (\mathbb{T})$, and $f(x+\frac{2\pi}{k})=f(x)$ for some natural number $k$. I want to show that $f$'s Fourier transform gets vanished for $n=rk+d$ where $1\leq d<k$. So here's ...
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141 views

Should I get the absolute value of the result of the inverse discrete fourier transform?

The result of equation 36 can be positive and negative.And if I don't get the absolute value of it,the ocean surface tend to be very regular.But according to the paper,the author never get the ...
4
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1answer
211 views

linear algebra approach to discrete cosine transform

I understand that the discrete Fourier transform simply changes basis to the discrete Fourier basis, which is an orthonormal basis of eigenvectors for any shift-invariant linear operator on $\mathbb ...
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1answer
507 views

Fourier Transform of a Homogeneous Heat Equation with a Source

Consider the heat equation $$\color{blue}{\begin{align} u_t&=ku_{xx}-bt^2u,\quad-\infty<x<\infty,\quad t>0,\\ u(x,0)&=\exp\left[-x^2\right]. \end{align}}$$ I am asked to solve it ...
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117 views

Bounding derivative of a function

Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, |f|>f_{max}$. Can we have a upper bound on the following, ...
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2answers
804 views

Fourier Transform Confusion

According to Wikipedia, the Fourier transform $\hat f$ of an integrable function $f:\mathbb{R}\to\mathbb{C}$ is defined by $$ \hat f\left(\xi\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi ...
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1answer
434 views

Complex results in inverse Fourier transform for simulating ocean water

I don't understand the equation37 in simulate ocean water by Jerry Tessendorf.The result is all complex number, how to be the slope.Even if I compute the magnitude of it,the result is just positive ...
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217 views

Dirichlet Problem: Uniqueness of solution

Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$. Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$? ...
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934 views

How can I prove that the Gibbs phenomenon overshoot for a Fourier Series is approximately 9%?

The question is pretty self explanatory, I'm studying Fourier Series with the book Mathematical Methods for Physicists written by Arfken and it does not explain that.
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1answer
102 views

Fourier Analysis on infinite groups

Is there something called (Fourier Analysis on infinite groups)? I have read some articles in Fourier analysis on finite groups but I wounder if there is such a theory on infinite groups!
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3answers
816 views

Fourier Transform of Dirac Delta Function

Dirac's delta function represents a wave whose amplitude goes to infinity as its duration in time goes to zero. It is a pulse of infinite intensity but infinitesmal duration. Please provide an exact ...
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421 views

Understanding Fourier Transform and FFT

First off, I'm sorry if this is a repost. I am currently writing my thesis, and I've been thrown into some Fourier analysis, which I know nothing of. So, even if this question has been posted before, ...
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75 views

Compute the number of zero of $f(z)=(z-1)(z-2)$ inside $C= \{|z|=3\}$ using the argument principal.

Compute the number of zero of $f(z)=(z-1)(z-2)$ inside $C= \{|z|=3\}$ using the argument principal. I am not sure what is the difference between a pole and a zero. I know the pole is where the ...
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0answers
254 views

Questions about the Fourier expansion of $e^{iz\cot(x)}$

By analogy with Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ ...
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1answer
501 views

Faster than Fast Fourier Transform?

Is it possible to make an algorithm faster than the fast Fourier transform to calculate the discrete Fourier transform (is there proofs for or against it)? OR, a one that only approximates the ...
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2k views

Fourier transform of fourier transform?

I have the definition of Fourier transform $$\hat f(\lambda) = \int_{\infty}^\infty f(t) \exp(- i \lambda t) dt$$ and have proved the following lemmas: $\hat E(x) = \sqrt{2 \pi} E(x)$ where $E(x) = ...
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1answer
101 views

Dirichlet problem: Obtaining the harmonic measure through Riesz representation theorem

For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$ $$ \Delta u=0, \text{ on } D, \\ \left. u\right|_{\partial D}=f \in C\left( \partial D\right). $$ With a fix $x$ in $D$, an ...
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1answer
139 views

Dirichlet problem: Is the Poisson Integral always a solution?

Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$. Is the Poisson integral of $f$, $$ Pf(x)=\int_{\partial D} f(t) ...
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822 views

Recursive Integration over Piecewise Polynomials: Closed form?

Is there a closed form to the following recursive integration? $$ f_0(x) = \begin{cases} 1/2 & |x|<1 \\ 0 & |x|\geq1 \end{cases} \\ f_n(x) = ...
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1answer
994 views

Finding the Fourier transform of $f(x) = \frac{a}{\pi} \frac{1}{a^2 + x^2}$ with the residue theorem

I keep getting the wrong answer for this problem! Find the Fourier transform of $f(x) = \frac{a}{\pi} \frac{1}{a^2 + x^2}$ using the residue theorem. Well, by definition: $$\hat f(k) = ...
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1answer
109 views

Differential wave like equation

Let $g$ be a twice differentiable function, set $$f = g''(x)$$ and assume that both $\displaystyle\int_\mathbb{R} |f(x)| \mathrm{d}x<\infty$ and $\displaystyle\int_\mathbb{R} |g(x)| ...
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1answer
91 views

How similar are the measures if their Fourier transform coincide?

Let $\mu$ and $\nu$ be finite Borel nonnegative measures exponentially decreasing at infinity, i.e. there exists $A > 0$ such that $$ \int\limits_{0}^{\infty} e^{Ax} \mu(dx) < \infty, \;\;\; ...
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1answer
66 views

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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293 views

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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74 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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1answer
83 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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72 views

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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73 views

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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2answers
241 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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1answer
506 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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216 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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1answer
236 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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1answer
137 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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1answer
737 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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1answer
104 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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471 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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1answer
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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1answer
992 views

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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1answer
85 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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113 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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1answer
104 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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1answer
236 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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96 views

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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1answer
68 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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1answer
766 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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1answer
214 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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245 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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474 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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2answers
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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1answer
144 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?