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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Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
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questions regarding Huseynov’s 2009 preprint titled ”On a class of entire functions, all the zeros of which are real”.

I bumped into this 2009 preprint by Huseynov titled ”On a class of entire functions, all the zeros of which are real” when I was reading Prof. Terry Tao's blog on "Tate’s proof of the functional ...
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Avoiding FFTs by reusing prior FFT results

Background From a mathematical point of view, the formulas similar to the following were produced: $F_1(f) = \mathcal{F}(T(t))$ $F_2(f) = \mathcal{F}(T(t)\times sin\Theta t)$ $F_3(f) = ...
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Where is the symmetry of Fourier transform in its implementation in Maxima and Wolfram Alpha?

From Wikipedia I saw that there is a symmetry of the Fourier transformation $F(F(f))(x) = f(-x)$ This matches the graphical explanation of the (German) Youtube video (9:15 to 9:45). I tried to see ...
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1answer
19 views

How to show convergence in $\mathcal{S'}(\mathbb R^{d})$?

We put, $\mathcal{S}(\mathbb R^{d})=$ The Schwartz space and $\mathcal{S'}(\mathbb R^{d})=$ The dual of $\mathcal{S}(\mathbb R^{d})$(The space of tempered distributions). Suppose $\alpha > 1$ and ...
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38 views

Understanding the Quantum Fourier Transform

I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. ...
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31 views

C++ code to compute 3d convolution [on hold]

I'm looking for a C++ package to perform 3d convolution between a 3d volume (for example, of size 384x13x13) and a 3d filter (for example of size 384x3x3). I tried googling, but found nothing ...
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1answer
65 views

Using Fourier Transform to solve heat equation

the heat equation of $U(x,t)$ on $-\infty<x<+\infty$ and $t>0$ is $$U_t=U_{xx}+\exp\left({\frac{-x^2}{2}}\right)$$ where $$U(x,t)\rightarrow 0 \quad as\quad x\rightarrow\pm\infty$$ and the ...
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1answer
31 views

Positive-definite + continuous at 0 $\Rightarrow$ continuous?

Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$. Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$? ...
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1answer
29 views

Fourier transform of $F(x)=\exp(-x^2/(2 \sigma^2))$

I am looking for the fourier transform of $$F(x)=\exp\left(\frac{-x^2}{2a^2}\right)$$ where over $$-\infty<x<+\infty$$ I tried by definition $$f(u)={\int_{-\infty}^{+\infty} ...
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Is DCT itself a Image compression technique or it just divides an image in 8*8 non overlapping blocks? [closed]

I was presenting my M-Tech thesis and during the presentation i was asked a question. My research work is based on removing blocking artifacts which comes usually after an image is compressed. During ...
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Is there a way to characterize the range of a chebyshev series through its coefficients?

Let $f$ be a Chebyshev series of order $n$ $$ f(x) = \sum_{i=0}^n a_i \cos\left( i \arccos\left(x\right)\right), x \in \left[-1, 1\right]. $$ Is it possible to characterize all the $\lbrace a_i ...
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36 views

Is 2D FFT separable?

Suppose I have a 2D matrix (or image). Can I loop on the columns - compute the FFT of each column and then loop on the rows (of the result matrix) and compute the FFT of that? Would that be equivalent ...
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Possion integral for measure dominated by the maximal function of the measure [closed]

Recently, I was thinking a problem about Possion integral for measure dominated by the maximal function of the measure, that is to say, let $\mu$ be a regular Borel measure in $\mathbb{R}$, define its ...
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1answer
22 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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1answer
26 views

Fourier Transform Properties Proof

If: $y(t) = x(t)*h(t)$ and $g(t) = x(9t)*h(9t)$ (Where * is convolution) How can I use properties of the Fourier transform to show: $g(t) = Ay(Bt)$ and find constants? I think A should be $1/81$ ...
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26 views

Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
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+50

Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs: $$\hat{L}u(x,t)=0$$ with the constraint $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$. During evolution, the ...
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How to find the global minimum of a function from its Fourier transformed function.

i.e Can $\min{f(t)}$ be expressed by $F(\omega)$? I have a series of data in frequency space. I can do discrete Fourier transform to time space to find its minimum. But I am wondering if there is a ...
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1answer
26 views

Fast Fourier Transformation: inverse transform of the product of polynomials

I have managed to implement and understand most of the Fast Fourier Transformation. However, I have one last question. If one has two polynomials, say $A(x)$ and $B(x)$, and one computes DFT of ...
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1answer
31 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
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Fast Fourier Transformation, explanation of $A(x)=A_0(x^2)+x A_1(x^2)$

I was looking at the Fast-Fourier Transformation today, on this site [if you cannot read Russian, simply use Google Translate, which is what I am doing right now]. http://e-maxx.ru/algo/fft_multiply ...
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56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
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1answer
32 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
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59 views

Equidistribution of $an^\sigma$

I am stuck at an exercise in Stein's book: Fourier Analysis, and it's exercise 8 in chapter 4. Show that for any $a\ne0$, and $\sigma$ with $0<\sigma<1$, the sequence $an^\sigma$ is ...
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1answer
71 views

Functions for which $\mathcal{F}g = f \ast f$

Suppose one is given $f \in L^{2}(\mathbb{R})$, my question is whether or not there exists a $g \in L^{1}(\mathbb{R})$ such that $f \ast f = \mathcal{F}g$ where $\mathcal{F}$ is the Fourier transform. ...
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Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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176 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
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1answer
45 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
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1answer
24 views

Is it true that, $x\rho(x/t)\in H^{s}$ for $\rho\in \mathcal{D}(\mathbb R)$ and $s>3/2$?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support; and the Sobolev space $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} ...
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$f\in L^{\infty}(\mathbb{R})$ as Fourier transform

i need to know if one can view a function $f\in L^{\infty}(\mathbb{R})$ as a Fourier transform of a certain function, say g? If the answer is positive please state the proof, or help me find one. ...
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4answers
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How to determine the periods of a periodic function?

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period ...
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Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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Intuition about generating functions

I am trying to gain some intuition about moment generating functions. In particular, for a random variable $X$, we have $$ \newcommand{\E}[1]{\mathbf{E}\!\left[#1\right]} M_X(t) = \E{e^{Xt}} = ...
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Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks
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Fourier Inversion transform

Is what i have highlighted in green a typo? Should it be $\pi e^{-|\xi|}$?
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26 views

Fourier transform of a triangular pulse

So I've been practicing some fourier transform questions and stumbled on this one; To start off, i defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as ...
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20 views

pFourier Transform using symbolic toolbox

Hi i need to use the symbolic toolbox from matlab and solve the following by fourier transform: $\int^{\infty}_{-\infty}\frac{\sin(at)\sin(bt)}{t^2}dt\text{ }\text{}\text{}$ So far the algo i came ...
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1answer
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Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
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characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
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1answer
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Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
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1answer
25 views

Paley-Wiener theorem for a sector $\{\zeta:-\epsilon<\arg(\zeta)<\pi+\epsilon\}$

One of the variations of the Paley-Wiener theorem yields: If $f\in L^2(\mathbb R_+)$, then the Fourier transform $F$, defined by $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ is a holomorphic function ...
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Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor, this question is kinda all over the place. If you have a set $B $ of $ N $ basis functions $ g_0(t), g_1(t), g_2(t), \dots, g_{N-1}(t) $ which are orthogonal over $[t_1, ...
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Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
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Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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104 views

Amount of sound near a specific frequency at a specific time

I have a sound signal, sampled at 48000 hz. Now I want to know 'how much sound' there is of a specific frequency (or near that frequency) at a specific time. For example, I want to know at 10s how ...
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Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
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What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...