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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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 ...
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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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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 ...
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1answer
40 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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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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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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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 ...
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42 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 ...
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76 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 ...
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30 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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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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11 views

How do I do harmonic analysis of a data set? [on hold]

I have data that looks like an arbitrary periodic wave. I want to calculate the coefficients of the appropriate harmonic series
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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 ...
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28 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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29 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 ...
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1answer
45 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 ...
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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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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
48 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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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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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\{ ...
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1answer
38 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} ...
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23 views

Function of bounded Fourier degree; bounding on subinterval

Suppose $f(x)=\sum_{|k|\le d} a_ke^{2\pi i kx}$, and it is given that $|f(x)|\le 1$ on $[0,L]$. Over all such functions, what is the maximum possible value of $$\max_{x\in [0,1]} |f(x)|?$$ (For ...
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+100

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} ...
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Fourier series qn determine the fourier series coefficients

Can someone please help me with this Fourier series $q_n$: determine the fourier series coefficients of $x(t)$ given as $x(t) = \cos4t + \sin8t+3$?
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Clarifying the Fourier Transform of $f_c(x)=\exp(-cx^2)$.

I believe I have found the Fourier transform of $f_c(x)=\exp(-cx^2)$ (where $0<c<\infty$) by noting first that $f_c'(x)=-2cx\exp(-cx^2)$. Taking the Fourier transform of both sides of this ...
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1answer
36 views

Construction bump function with positive Fourier transform

I am looking for the construction of a smooth bump function, $f$, mapping the real line to itself which has two special properties: (1) $f$ is constant on some interval in its support (for instance ...
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29 views

Fourier transform on trig wave

Find the fourier transform for signal in this picture (sorry for the bad quality) Could it be done like this? The signal is a sum of two triangular waves that are each delayed. ...
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Extensions to higher dimensions by tensorization. Unitary DFT in 2D?

I have problem understanding the underlying concept of tensoration (if there is such term). Fist of all the unitary DFT is NxN. Is it 1D ? How does it look when we increase the dimension let say to ...
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43 views

Continuity in the complex plane

I was reading a book where it is claimed that a sufficient condition for \begin{equation} f(x)=\frac{1}{2\pi}\left|\sum_{j=0}^{\infty}\theta_je^{ix j}\right|^2 \end{equation} to be continuous and is ...
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Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
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Understanding JPEG compression.

I have some problems in understanding a passage of the JPEG compression algorithm: Consider an $8\times8$ matrix $M$ that in our case is a "piece'' of a channel (for example the red channel $R$) of ...
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39 views

Fourier Transform of Heaviside Function

I'm trying to find the Fourier transform of $H(k - |x|)$, where $H$ is the Heaviside step function. I've solved a few Fourier transforms recently, but this one is giving me a bit of trouble. I'd ...
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1answer
33 views

exponential term evaluation doesn't make sense in this example

I am studying for my final and doing some practice questions, but I am confused by something: Here the solution says k at 0 we get N/2, but there is no way that answer is correct. If k is at 0 the ...
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Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why?

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why ?
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Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
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1answer
29 views

Expression for characteristic function of a truncated RV

Let $\langle\Omega,\mathscr{F},\mathbb{P}\rangle$ be a probability space, let $X$ be a random variable defined thereon with density $f$ and $\phi$ be its characteristic function. Then if $A \in ...
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Using Fourier Analysis to determine Green's Function of Laplace's equation

I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have ...
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25 views

Fourier Transform of a function with sinusoidal sampling

What is the relation between the Fourier Transform (FT) of $f(x)$ with regular sampling and the FT of $f(x)$ with sinusoidal sampling? In other words, it's a FT of a function composition $f\circ ...
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Support of polynomial distributions

Assume $u\in\mathcal{S}'(\mathbb{R}^n)$ is a tempered distribution such that $\widehat{u}$ is compactly supported and $u^k$ defines a distribution for each $k=1,\cdots,m$. Let $p_1,\cdots,p_m$ be ...
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1answer
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Fourier transformation of h(-t)

Ask a simple question: we know $F[h(t)] = H(f)$, where $h(t)$ is the impulse response. How to show $F[h(t)] = H^*(f)$? My answer is just $H(-f)$.
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39 views

Fourier Transform of $1/(\pi\cdot t)$ by Duality

I'm asked to prove using "duality property" the Fourier transform of $$\frac{1}{\pi t} = -j sgn(f)$$ I have the proof steps but I'm quit not understanding it: multiply by $j = \frac{j}{j(\pi t)}$ ...
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1answer
27 views

Fourier transform Excercise

I am stuck on an excerise which says that prove the fourier transform $f(k)$ of a real function satisfied the condition $f(-k)=f*(-k)$. Where the astericks denotes the complex congugate. I am ...
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1answer
24 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period ...
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1answer
36 views

Rewriting $e^{-a|t|}$

here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this ...