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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two dimensional extension of the Fourier transform of $|x|^\alpha$?

I know the Fourier transform pair \begin{equation} |x|^\alpha \leftrightarrow -2\sin(\pi\alpha/2)\Gamma(1+\alpha)|\omega|^{-1-\alpha} \end{equation} Can this formula be extended to two dimensional ...
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1answer
11 views

Fourier coefficients, Fourier spectrum and signal energy

I was wondering if the equation 1) is correct in this example. The author described the difference between equations for $X(w)$ and $X[k]$. The only difference I see is the $\frac{1}{T}$ coefficient. ...
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2answers
35 views

What is the realtionship between a cosine function and a Dirac delta? [on hold]

I want to compute the Fourier transformation of a cosine function $\cos(2\pi f_0 t)$, but I don't know if there are any relationships between trigonometric functions and Dirac delta's.
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1answer
36 views

$\int_{0}^{2\pi} f(x) \text{e}^{-ilx} \, \text{d}x =0$

Suppose $f,g \in \mathcal{L}^1(\mathbb{R} / 2 \pi)$ with $f(x)=g(mx),m \in \mathbb{Z}$. I want to show that $$ \forall \, l \in \mathbb{Z} \ \text{with} \ l \not\equiv 0 \ \text{mod} \ m \colon ...
2
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1answer
34 views

Fourier transform of a divergent function

During a calculation of Feynman diagram, I encountered an integral which is diverging: $$\int d^{2}p\frac{p^{2}p_{n}}{\left(p^{2}+\alpha k^{2}\right)^{2}}e^{ip\cdot y}$$ where $p$, $k$ and $y$ are ...
1
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1answer
30 views

Does convolution of two functions in $H^s(\mathbb{R})$ belong to $H^{2s}(\mathbb{R})$?

Let $f$, $g$ be two density functions and assume that $f,g\in H^s(\mathbb{R})$, $s>\frac{1}{2}$, where $${H^s}(\mathbb{R}) = \left\{ {u:\int_{ - \infty }^\infty {{{\left| {\hat u\left( t \right)} ...
2
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1answer
29 views

Expanding in a Fourier series $y = |\cos x|$

How to expanding in a Fourier series function $y = |\cos x|$? Especially interested in how to find $$a_n= \frac{2}{\pi}\int\limits_{0}^{\pi}|\cos x|\cos(nx)dx$$
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1answer
31 views

Jump discontinuity of a function and its analytic phase

Let $f:\mathbb{R}\to \mathbb{R}$ and $f \in BV(0,1)$ with a jump discontinuity at $x = a\in(0,1)$. Let $f_h$ be its Hilbert transform and let $$f_A(x) = f(x) + i f_h(x)$$ Is it true that the function ...
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0answers
18 views

Is this a viable generalization of Newton series?

I wonder if the following formula a viable generalization of Newton's series. $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^x \int_{-\infty}^{+\infty}e^{i\omega t}\sum_{m=0}^\infty ...
2
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2answers
41 views

Evaluate fourier coefficient of $f(t)=t$.

Evaluate the Fourier coefficient of $f(t)=t$. $$\hat{f}(n) = \frac{1}{2\pi}\int_0^{2\pi} te^{-int}dt$$ I'd be glad for help with this calculation. My integration skills need an improvement. My ...
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0answers
20 views

Integral of complex exponential distribution [on hold]

I have to prove, that $$\forall a\in\mathbb{R}^n:\int_{\mathbb{R}^n}e^{-2\pi i(a,x)}\varphi(x)\mathrm{d}x=\varphi(a)$$ or more precisely $T_{e^{-2\pi i(a,x)}}=\delta_a$ where ...
1
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2answers
163 views

How one can implement the equation with “$i$” in it?

I have an equation: $$f(t)=c(e^{i2\pi\frac{n}{T}t}+e^{-i2\pi\frac{n}{T}t})$$ ...for $t\in(-\pi,\pi)$, and with $T=2\pi$. I have to draw a plot of the function $f(t)$ for $n\in\left \{0,1,2,5 \right ...
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1answer
18 views

Help understanding the output from Apache Commons Math's Discrete Fourier Transform

I'm using a discrete Fourier transform to translate a finite set of samples to the frequency domain. I'm trying to start with a very simple set, but am still getting confused. I'm starting with this ...
2
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0answers
30 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
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0answers
19 views

Fast Fourier transfrom

What are the prerequisites for understanding the fast fourier transform for fast multiplication? What topics should I be familiar with first?
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29 views

DFT of vector $(0, 1, 2, 3)$

The problem is that my answer is different from answer i get in MATLAB. My answer is $(6, -2-2i, -2, -2+2i)$ while MATLAB answer is $(6, -2+2i, -2, -2-2i).$ In MATLAB i use command ...
2
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1answer
37 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
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0answers
14 views

Where can I find material on polynomial filters?

Most students and mathematicians probably know a fair amount on roots-of-unity filters, or on Fourier analysis. The basic notion of this "filtering" is, given a polynomial, we can find the $n$th ...
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55 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
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1answer
35 views

Why is a wave with high FM aperiodic?

I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When ...
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3answers
49 views

Is the Fourier series a “linear transform”?

Fourier series fundamentally involve the sine and cosine functions: $$a_0+\sum_{k=1}^\infty \left(a_k \cos kx+b_k \sin kx\right)$$ These functions are about as non-linear as you can get. But... is ...
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1answer
25 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
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0answers
18 views

Fourier transform of $be^{i k y^b}/y^{1-b}$

I'm trying to compute the Fourier transform of $$ \frac{ be^{i k y^b}}{y^{1-b}}$$, i.e. $$ F(z) = \int_{-\infty}^\infty \frac{ be^{i k y^b}}{y^{1-b}} e^{i z y}dy$$ I tried using Mathematica for ...
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0answers
64 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
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1answer
8 views

2D DFT of Hexagonally Sampled Grid

Is there a good way to perform a 2D FFT of discrete data on a hexagonally sampled grid? The best method I've got so far involves oversampling the hex grid to a rectangular grid, and performing the 2D ...
0
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1answer
37 views

Convergence of a sequence involving integral

Consider $f:[-\pi, \pi] \to \mathbb{C}$ is analytic (infinite differentiable) and periodic. Define $a_n:= \frac{1}{2 \pi}\int_{-\pi}^\pi f(x) e^{-inx}dx$ (the Fourier coefficient of $f$). Show ...
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28 views

Exercise 22, Chapter 5 of Stein and Shakarchi's Fourier Analysis

I am working through Stein and Shakarchi's Fourier Analysis and am stuck on this exercise. I would appreciate any hints. The statement of the exercise follows. Notation: $\mathcal{S}$ is the Schwartz ...
2
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1answer
41 views

Periodicity of an infinitely differentiable function

Consider $f:[-\pi,\pi] \to \mathbb{C}$ be an infinitely differentiable function with $f^{(n)}(-\pi) = f^{(n)}(\pi)$ for all $n \in \mathbb{Z}^+$. Is this a periodic function ? I think it is a ...
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0answers
36 views

How to integrate this 1D Fourier transform?

\begin{equation} \int _{-\infty} ^\infty |s|^{-5}~e^{-(s-s_0)^{-4}} e^{\imath st}ds \end{equation} where $s_0$ is a positive real number
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1answer
29 views

Schwartz space on $\mathbb T^{n}$

For the definition of Schwartz space space on $\mathbb R^{n},$ see this. My Questions: (1)Is it make sense to talk of Schwartz space on torus $\mathbb T^{n}$ ? If yes, what can be the analogous ...
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13 views

Calculate FFT of 1/r green's function

I am trying to write the Poisson equation solver in C, using FFTW library. For given density of charge I need to calculate potential assuming periodic boundaries. My idea is to use convolution, simply ...
2
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45 views

A deep understanding of the Fourier transform

I feel like i don't understand the Fourier transform. I've seen what it does and its properties but even after reviewing various proofs i don't understand why we end up explicitly with a relation ...
2
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0answers
81 views

how to use Matlab ifft to calculate the following integral? [duplicate]

$$R(t)=\int_{-\infty}^\infty\dfrac{\omega e^{i\omega t}}{(3-\omega^2)^{2}+4\omega^2}\,d\omega$$ where t is a integer and $t>0$ I used to calculate this integral by numerical integral,but it seems ...
2
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1answer
31 views

The Fourier transform of $ e^{-|x|^\alpha}, \alpha>0. $

Do you know the Fourier transform of $$ e^{-|x|^\alpha}, \alpha>0. $$ Does it have an implicit formula. In the spacial cases $$ (e^{-|x|})^\hat{\,}(\xi)=\frac{2}{1+4\pi^2\xi^2},\ ...
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0answers
9 views

Inverse Fourier Transform of $1/k^2$ in $\mathbb{R}^N $

This comes up in the context of finding the Green's function of Poisson's equation for $\mathbf{x} \in \mathbb{R}^n $ $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Attempt by using Fourier ...
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0answers
17 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
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1answer
20 views

control of an integral using maximal function

Let $I$ be a compact interval with center $c(I)$ and N be a large positive integer. It seems to me that there exists a constant $C$ such that for any good function $f$ (e.g. Schwartz function) we have ...
4
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0answers
33 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
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0answers
30 views

How to prove $\hat f$ is uniformly continuous in $R^n$?

Let the Fourier transform be defined by $\hat f(\xi)=\int_{R^n}f(x)e^{-ix\xi}dx$. Suppose $f\in L^1(R^n)$. How to prove $\hat f$ is uniformly continuous in $R^n$?
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1answer
14 views

Using partial fraction decomposition to find inverse Fourier transform

I've reduced my problem to $H(w) = \dfrac{1}{(1-\frac{1}{4}e^{-jw})(1-\frac{1}{3}e^{-jw})}$. I need its inverse discrete Fourier transform. My thinking is that I could use partial fraction ...
4
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1answer
46 views

When is an oscillating integral small?

I hope, the title is not too confusing. My question is the following: We all know the Riemann-Lebesgue-Lemma stating that for $f\in L^1(\mathbb R)$, one has $$ \lim_{k\to\infty} \int ...
2
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0answers
22 views

An $f\in H^{1/2}$ with self-convolution, showing it is an $C^1$ function.

If $f\in H^{\frac{1}{2}}(\mathbb{R})$ is a Sobelev 1/2 function that $f=f*f$, then how do you show that $f\in C^1$ with a bounded derivative.
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17 views

Plancherel's theorem variants

How would you prove a variant form of Plancherel theorem: If $(c_n)_{n\in\mathbb{Z}}$ are coefficients and $\sum_{n\in\mathbb{Z}}|c_n|^2<\infty$, then there exists a unique function $g\in L^2(0,1)$ ...
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1answer
23 views

Relating the Fourier transform of two functions.

We are given that $f\in L^1(\mathbb{R}^k)$ and that $A$ is a linear operator on $\mathbb{R}^k$ (I'm assuming that $A:\mathbb{R}^k\to\mathbb{R}^1$, but correct me if that is incorrect). We also have ...
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1answer
20 views

Fourier transform of a function with sine [duplicate]

I don't know how to compute the Fourier tranform of this function: $f(x) = \frac{\sin \pi a x}{\pi x}$ I know that $\frac{\sin \pi a x}{\pi x} = \frac{e^{i \pi a x} - e^{- i \pi a x}}{2i \pi x}$ ...
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0answers
11 views

different way to compute power spectral density

I am writting a piece of code to compute power spectral density (psd) of a signal and wanted to compare two approaches : compute the FFT of the signal and square its amplitude compute the biased ...
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1answer
49 views

What is the Fourier transform of $\frac{x}{\sin(x)}$?

What is the Fourier transform of $\frac{x}{\sin(x)}$? (Not $\frac{\sin(x)}{x}$!)
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0answers
31 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
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14 views

Solution of a differential equation with problem of Cauchy

The question is the next: What can I say from the existence, uniqueness and continuos dependence of the solution? Is this a strongly continuos one-parameter group or a semigroup. $ \left\{ ...
2
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1answer
39 views

Square root of a Fourier series

This problem came to mind in conjunction with two earlier ones [1] [2]. Let $f(x)$ be positive square-integrable function on $[0,2\pi]$ with Fourier series $\sum\limits_{n=-\infty}^{\infty} ...