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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Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
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1answer
33 views

Norm convergence of approximations to the identity

Let $\varphi \in L^1(\mathbb{R}^d)$ be such that $$\int_{\mathbb{R}^d} \varphi(x) \, dx = 1.$$ For each $\varepsilon>0$, let $\varphi_\varepsilon:= \varepsilon^{-d} \varphi\left( \dfrac x ...
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0answers
58 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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1answer
6 views

Increasing order of fourier coefficients on the boolean cube

Given a function $f:\{0,1\}^n\rightarrow \{0,1\}$, is it true that for any $S,T\subseteq[n]$, such that $S\cap T =\phi$, then $\hat{f}(S\cup T)\leq \hat{f}(S)$? It seems so to me cause, if if you just ...
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0answers
14 views

How to introduce a new equivalent using two separate boxes in variables?

I am reading a paper in mathematics about Fourier Transform. It consider two boxes A and B with centers of $ x_0(A) $ and $p_0(B)$ respectively. It introduce a new function $R^{AB}(x,p)$ as follows: ...
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1answer
59 views

Inter-neighbor resistance on triangular prism

Given a triangular prism of infinite length along the X direction. A graph is formed with the set of nodes all the points on an edge of the prism with integer values of X, and the with each node ...
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1answer
34 views

Representation of Heaviside function's Fourier transform

I've seen here that the Fourier transform of Heaviside function $\Theta(t)$ is $$ \Theta(\omega) = \frac{1}{i\omega} + \pi \delta(\omega) \tag{1}$$ But in some physics texts and here I've seen the ...
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1answer
47 views

What if the Fourier series of a periodic function also has periodic coefficients $a_k$

If given that $x(t)$ is a periodic continuous time signal, with periodic $T$. It can be expressed by the Fourier series, i.e. $x(t)=\sum\limits_{k=-\infty}^{+\infty}\,a_k\cdot e^{j k \frac{2 ...
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0answers
23 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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1answer
46 views

Fourier transform of cosine with square root

In relativistic mechanics, i came across the Fourier transform of the following function : $\cos \left(t \sqrt{x^2+m^2} \right)$ or $e^{it \sqrt{x^2+m^2}}$ ($t$ and $m$ are constants). Is there a way ...
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1answer
43 views

Solving wave equation by fourier method

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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1answer
26 views

When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
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4answers
67 views

Dirac Delta function inverse Fourier transform

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
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0answers
26 views

Dirac Delta function inverse Fourier transform [duplicate]

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
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1answer
18 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
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1answer
37 views

An equality about Fourier transform

I have read an equality about Fourier transforms which I can not proof. It is as following: Let $u\in C_0(\mathbb{R}^n)$ and \begin{equation} g(x_1,x_2,...,x_{n-1}):=u(x_1,x_2,...,x_{n-1},0). ...
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1answer
40 views

Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
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1answer
27 views

In what sense is the Schwarz class of functions a “class”?

As an engineer who has not learned very much modern algebra, I recently learned about "class" in the algebra sense. Then I remembered our professor calling the set of Fourier transformable functions ...
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1answer
33 views

Fourier analysis notation - Sh and Ch

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!
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1answer
66 views

Fourier transform is real if $f$

I want to prove that the Fourier transform $F(\xi)$ of a function $f$ will be a real function when, and only when, $f(x)$ is an even function. I'm using the following definition of Fourier ...
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1answer
41 views

A Hölder continuous function whose Fourier coefficients do not decay very fast

At Stein's book of Fourier analysis (Chapter 3, page 91, exercise 15) I was trying to solve the following problem I have to prove that the result ...
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1answer
67 views

What is $\lim_{n \to \infty} n^3 a_n$? [duplicate]

$a_n$ is the Fourier coefficient of $$f(x) = \left(1 - \frac{|x|}{\pi}\right)^4$$ The answer is infinity, but can someone give an answer that doesn't require explicit computation of the $a_n$? I'm ...
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1answer
19 views

Bessel equation of half-order (asymptotic)

Not really optimistic about getting a reply for a question tagged under "Bessel function" but here goes, I have $$J_{\frac{1}{2}} = (a_1 \cos(z) + a_2 \sin(z))Z^{-\frac{1}{2}} $$ and ...
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1answer
81 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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1answer
16 views

1D FFT on rotated image column by column

I am facing a problem: performing 1D FFT on a rotated column by column on a rotated image, described as following: Original Image: Rotated Image: What I have: original image convolution ...
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1answer
428 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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1answer
50 views

Finding the limit that involves Fourier coefficients,

Given the function $f(x) = 1 - \dfrac{|x|}{\pi}$, I had computed its Fourier coefficients, using integration by parts and got: $$ a_n = \begin{cases} 0, & \text{for $n$ even}, \\[6pt] ...
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1answer
62 views

Central Limit Theorem proof: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
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27 views

Central Limit theorem: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected [duplicate]

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
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76 views

Hausdorff-Young / Restriction Inequality

Let $\lambda$ denote Lebesgue measure on $\mathbb{R}^d$. The Hausdoff-Young inequality is that $$ \| \widehat{f} \|_{L^{q}(\lambda)} \leq \| f \|_{L^{p}(\lambda)}. $$ when $1 \leq p \leq 2$ and ...
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When would a Fourier Product (made up term) exist for a finite sequence of the form $C_{\text{Max}}\prod _{i=1}^k A_i \cos \left( B_i n\right)$

Let us say that we are given a finite list of points of the form C = {i,$x_i$} where i goes from 0 to the card(C) that when plotted in the Euclidean plane has some vertical axis that splits the graph ...
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12 views

The bond between Fourier Transform and Epicycle theory

Can someone help me understanding the bond between the Fourier Transform and the epicycle theory? I have searched in many places such as: http://math.stackexchange.com/a/72479/185138 ...
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1answer
36 views

Fourier transform division theorem in $\mathbb R^n$

It is known that if $f \in L^1(\mathbb R)$, $\widehat f(\xi) \neq 0$ for any $\xi \in \mathbb R$, then for any $h \in L^1(\mathbb R)$ such that $\widehat h$ is compactly supported there exists $g \in ...
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1answer
13 views

Simplification of this fourier transform signum function

Given this equation: $$\frac{-1}{4c}[\int_{ -\infty}^{\infty}g(\varpi)Sgn(x - ct - \varpi).d\varpi -\int_{-\infty}^{\infty}g(\varpi)Sgn(x+ct - \varpi ).d\varpi ]$$ Where sgn is the signum function, ...
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1answer
40 views

Fourier transform of a piecewise

How should I go about seeking the Fourier transform for the piecewise function: $$f(x) = \left\{\begin{matrix} 0 ,&|x|>a \\ 1 ,&|x|<a \end{matrix}\right.$$ Is this the correct ...
2
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0answers
15 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
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0answers
6 views

A theorem regarding epicycles

Can somebody help me understanding the theorem on the last page of that article about Fourier Series and Epicyles? ...
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1answer
14 views

Functions of polynomial growth and the Schwartz space

A smooth function $m \in \mathcal C^\infty(\mathbb R^n)$ is said to be slowly increasing if for all $\alpha \in \mathbb N^n_0$ there exists $C_\alpha, k_\alpha$ such that $|\partial_\alpha f(x)| \leq ...
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2answers
57 views

Fourier transform of the 1-d Coulomb potential

Though it may sound like a physical problem, but the thing I will introduce is rather mathematical. For the Fourier transform of Coulomb potential $$ V(\vec{x})=\frac{1}{\vert x\vert} $$ I can ...
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0answers
30 views

How condition for existence of Fourier transform is valid?

The condition for Discrete time Fourier transform to exist for function $f(n)$ is given as $$\sum_{-\infty}^\infty |f(n)| < \infty.$$ In case of continuous Fourier transform the difference is ...
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1answer
32 views

Fourier series and transform (epicycles)

Let $\gamma:[a,b]\to\mathbb{C}$ be a continuous curve. 1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times ...
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1answer
128 views

How to use Fourier Transform with non-trivial boundary conditions such as in potential flow around a plate?

I'd specifically like to be able to solve this PDE with boundary conditions corresponding to flow around a line (plate cross-section), otherwise known as flow-tangency, with integral transforms. ...
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1answer
481 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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3answers
191 views

A more elegant way to find the Fourier transform

Let $f$ be defined analytically as : $$f(x)=\arccos \left ( \sin \left ( 2x \right ) \right ), x \in\left (0,10 \right ], f(x)=0, x\notin\left ( 0,10 \right ]$$ Here is a graph of the above ...
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0answers
15 views

FFT differential equations

Given a generical differential equation what is the procedure to solve it using fft command. Can anyone explain me how to do it? For example: $$\frac{d^2y}{dt^2}+10\cdot \frac{d\:y}{dt}=-5\cdot ...
4
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1answer
295 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
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5answers
43 views

“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
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1answer
28 views

Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
2
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1answer
37 views

Proof of elementary Wiener's tauberian theorem

I want to proof the following (simple case/version of) Wiener's tauberian theorem: The (span of the) set of translates $\{f_a | a \in \mathbb R^n\}$, $f_a(x) = f(a+x)$, is dense in $L^2(\mathbb R^n)$ ...
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1answer
22 views

Fourier Transform existence

Let $\varphi:\mathbb{R}^2\to\mathbb{R}$ be a continuous function. Moreover, consider that $f:\mathbb{R}\to\mathbb{R}$ is a schwarzian function, i.e. $f\in C^{\infty}$ and $\lim\limits_{x\to\pm\infty} ...