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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How to extract power of top two frequencies of a spectrum without using an FFT?

What I'm trying to do is see if a particular frequency component becomes dominant (and I don't really know what the dominant frequency is). Therefore, I figured that I can get the top two peaks of ...
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1answer
63 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
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28 views

Operator $T$ commutes with all translations $\Leftrightarrow$ $T$ is multiplication operator

Let $T: L^2(\mathbb R) \to L^2(\mathbb R)$ be an operator that commutes with all translations $f(x) \mapsto f(x-y)$. Why does it follow that the Fourier transform of $Tf$ is then given by ...
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1answer
33 views

Fourier transform with initial limit starting from 0

For a function $f(p)$, the Fourier transform is $\hat{f}(p)=\int_{-\infty} ^{\infty}e^{-i xp}f(x)dx$. What are the conditions that i write it as; $\hat{f}(p)=2\int_{0} ^{\infty}e^{-ixp}f(x)dx$? I ...
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50 views

How to Prove Plancherel's Formula?

I have difficulty in proving Plancherel's formula in Fourier transform. Here is what I have thought: In this question, I denote complex conjugation by an overline and (inverse) Fourier transform is ...
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1answer
65 views

Fourier transform of the Heaviside function

As you can see from the title I want to calculate the Fourier transform of the Heaviside function $u(t)$. Proven the the Heaviside function is a tempered distribution I must evaluate: $$ \langle ...
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1answer
20 views

$\{f\in L^{1} : \hat{f} \in L^{p} \}$ closed under convolution?

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ We note that $L^{1}(\mathbb R) \ast ...
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98 views

Sets of Divergence for Fourier Partial Integals

It is a consequence of Carleson's theorem together with a transference argument that (see Section 4.3.5 in L Grafakos, Classical Fourier Analysis for proof) that the Fourier partial integrals of a ...
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1answer
31 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
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42 views

Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
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34 views

What can we say about the transform of a function on a truncated domain, with respect to the transform on the full domain?

Let $f$ be a function on R and $\hat f$ its Fourier transform. Consider a truncated version of $f$ called $\bar f$ whose value outside an interval is $0$. Formally, $\bar f(x) = f(x) * 1_{x \in I}$ ...
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1answer
31 views

Question about the limits of definite integrals

Let me take an example that I've come across while studying Fourier series, We all know that $$\int_{-a}^{a} \sin \left( \frac{n\pi x}{a} \right) dx = 2 \int_{0}^{a} \sin \left(\frac{n \pi x}{a} ...
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2answers
80 views

Role of i in Fourier transform

I've seen several derivations of the Fourier transform, but most don't cover the conversion to the form $$ S(f) = \int_{\infty}^{-\infty} s(t)e^{-i2\pi ft} \;\mathrm{d}t $$ What is the role of ...
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33 views

Show there exists $n\in \Bbb{Z}$ such that $\lambda =({2\pi n\over T })^4$.

Let $f:\Bbb{R}\to \Bbb{C}$ be an infinitely differentiable function with period T. Also, $f^{(4)}=\lambda f$. Show there exists $n\in \Bbb{Z}$ such that $\lambda =({2\pi n\over T })$. Attempt: I did ...
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34 views

The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is non-negative. We can observe that the function $|f|^2$ has a nice property : ...
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1answer
39 views

Integration of a generic radial function in polar coordinates

I need to perform the following integral $\int{P(k) e^{i \vec{k}\cdot \vec{\Delta r}} \frac{d^2k}{(2 \pi) ^2}}$ using polar coordinates. I think the result should depend on some Bessel function, but ...
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36 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
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2answers
36 views

Confused by certain interpretation of expected value…

I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density. "Consider the simpler (idealized) situation where we are ...
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35 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz [duplicate]

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
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60 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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1answer
90 views

Walsh/Hadamard/Fourier Transform

Hey guys can anyone explain to me what the Walsh/Hadamard/Fourier Transform actually does and how and when do I use it? Can you also recommend me some textbooks that I can use to help me understand it ...
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25 views

About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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62 views

Show there exist a constant $c\in \Bbb{C}$ such that $\int_{0}^{1}|{f-c}|^2<{1\over 36}$

Let $f:\Bbb{R}\to \Bbb{C}$ be a $1$-periodic function, $f\in C^1$ and $\int_{0}^{1}|f'|^2\le 1$. a. Show $\sum_{k\ne 0}|{\hat{f}(n)}|^2\le {1\over 4\pi^2}$ (I did it already, and that question is ...
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53 views

Does the Fourier series converge at $x=0$?

Let $f(x)$, a $2\pi$ periodic funciton such that $f(0) = 1$ and for every $0\ne x\in[-\pi,\pi]$: $f(x) = 1 + \sin \frac{\pi^2}{x}$. Is the Fourier series of $f(x)$ converges at $x=0$? If so, what ...
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3answers
59 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
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1answer
26 views

Euler equation formula

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$ I am getting following integrals: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
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39 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
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19 views

Applying the inversion formula to Hardy's theorem

So, in the proof of Hardy's theorem, who says that $\zeta$ has infinite zeros on the critical line, we have eventually that ...
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378 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
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38 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and ...
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175 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
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44 views

2D Discrete Fourier Transform on an Image - Example with numbers (rgb)

I am trying to write my own function that takes an image, an pixel by pixel it calculates that pixel value that will produce a 2D Fourier Transform image. I have no idea about signal processing, my ...
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54 views

Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
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46 views

What is the Fourier transform of Riemann Zeta function?

All: Is there an explicit form of Fourier Transform of Riemann Zeta function ? Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ? I remembered I had seen something like ...
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1answer
53 views

Fourier transform of exponent

I need to count the Fourier transform of the following function but it does not seem so obvious for me. $f(x)=(e^{-ab})-1$ for $x\ge0$ and $f(x)=0$ for $x<0$ where: $a=1$ and $b=-1$ I don't ...
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1answer
36 views

$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
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1answer
31 views

Gradient of the Fourier transform of a function

I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function. In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is ...
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1answer
56 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
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1answer
33 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
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259 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
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112 views

Fourier sine transform of $\frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$

Show that $$ \int_0^{\infty} kF(k)\sin(ka)\,dk = \frac{\pi}{2}aG(a) $$ where $$ F(x) = \frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert $$ and $$ G(x) = \frac{\sin x-x\cos x}{x^4} $$ EDIT: ...
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23 views

A convolution equation with two unknowns

I consider the following convolution-type equation with two unknowns $f_1$, $f_2$: $$ a_1 * f_1 + a_2 * f_2 = 0, $$ where $a_1$, $a_2 \in L^1(\mathbb R)$ and $*$ is the ordinary convolution. This ...
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1answer
77 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
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1answer
50 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$ [closed]

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to ...
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55 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
3
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55 views

On the Fourier transform of $f(x)=\ln(x^2+a^2)$

I would like to derive the Fourier transform of $f(x)=\ln(x^2+a^2)$, where $a\in \mathbb{R}^+$ by making use of the properties: \begin{equation} \mathcal{F}[f'(x)]=(ik)\hat{f}(k)\\ ...
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29 views

Fractional derivative of $e^{-x^2/2}$ using Fourier transform and Taylor series

I am not familiar with fractional calculus, so I want to know what I am doing wrong. The convention I use $$\int^\infty_{-\infty}e^{-\frac{x^2}{2}}e^{-i k x}dx=\sqrt{2 \pi}e^{-\frac{k^2}{2}}$$ I am ...
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1answer
33 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
2
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3answers
97 views

Prove that the following function is $C^{\infty}$ [duplicate]

Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}}, & \text{if $|x|<1$} \\ 0, & \text{if $|x| \ge 1$} \end{cases}$$ is $C^{\infty}$ I found this problem ...
3
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2answers
71 views

Replicating Kolmogorov's Counterexample for Fourier Series in Context of Fourier Transforms

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More ...