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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$||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||f^{(n)}||_1)^{1/n}$

Let $f \in L^2 \cap L^1$ on the Real line, and define $f^{(n)}$ to be the $n$-fold convolution $f \circ f ... \circ f $. I want to show that $||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||...
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34 views

Extension of Fourier transform to $L^2(\mathbb{R})$

We defined the fourier transform and it's inversion for the Schwartz class $S(\mathbb{R})$. Since $S(\mathbb{R})$ is dense in $L^2(\mathbb{R})$, we can find for a given $f\in L^2(\mathbb{R})$ a ...
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1answer
12 views

For a compact abelian group, what can I conclude if range of a character is finite?

Suppose $G$ is a compact abelian group, and suppose $<g^n>$ is dense in $G$ where $g$ is a particular element of $G$ and $<g^n>$ is the subgroup generated by $g$. Let $\chi$ be a character ...
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19 views

Fourier transform of convolution with additional dependence

There's the well-known identity $$\widehat{f*g}(\xi)=\hat f(\xi)\hat g(\xi)$$ for, say, $f,g\in\mathcal S$. Does anyone know of an extension of this to a situation like $$\mathcal F_x\{[f*g(\cdot,x)](...
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1answer
19 views

Square-integrability of Fourier transform

Given an absolutely integrable function $f(x)$, with continuous and absolutely integrable first and second derivatives, is it true that its Fourier transform $F(t)$ is square-integrable? I know that ...
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39 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
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Did Joseph Fourier ever make a pure mathematical mistake?

Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001: It is remarkable that the nineteenth century was a time of error for mathematics: not trivial oversights or amateur ...
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1answer
12 views

Prove $\mathscr{F}[(1+|x|^2)^{-s}]\in L^1(\mathbb{R}^d)$

Let $s>0$, show that $\mathscr{F}[(1+|x|^2)^{-s}]\in L^1(\mathbb{R}^d)$. The original goal is to prove that $W^{s,p}(\mathbb{R}^d)\hookrightarrow L^p(\mathbb{R}^d)$ for all $s>0,1\le p\le \...
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31 views

Is my formula for DFT correct?

I'm doing "Digital Image" online course. I tried to solve the following question $x(n_1,n_2)$ is defined as $x(n_1,n_2)=(−1)^{(n_1+n_2)}$ when $0≤n_1$, $n_2≤2$ and zero elsewhere. Denote by $X(k_1,k_2)...
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28 views

Compute two 1D lookup tables for computing DCT of a 2D Laplacian

I am following a paper that I have to implement through coding and I am struggling understanding what it means exactly with the sentence "create two 1D lookup tables for computing the DCT of a 2D ...
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23 views

Numerical method for fourier transform other than FFT/DFT

FFT relies on uniform samples, which cause aliasing, so FFT can be inaccurate in a certain case. Suppose you can obtain samples of $f(t): \mathbb{C} \to \mathbb{C}$ at any point ($t$ can also be a ...
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13 views

Convolution notation

I refer to notations like $$f*K_\epsilon(0)$$ in Convolution with Gaussian question. Do they mean $(f*K_\epsilon)(0)$, i.e. the convolution evaluated at zero or $f*(K_\epsilon(0))$, i.e. $f$ ...
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1answer
55 views

In a proof of the Riemann Lebesgue lemma

In a proof of the Riemann-Lebesgue lemma in Hunter's Applies Analysis, he first proves the statement in the Schwartz space and then uses a density argument: Here are my questions: What goes ...
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1answer
20 views

Short-time Fourier Transform identity in $L^2$

Define the Short-time (or windowed) Fourier Transform of a function $f:\mathbb{R}\rightarrow\mathbb{C}$ as follows, $F_gf(\omega,t)=\int\limits_{\mathbb{R}}f(x)\overline{g(x-t)e^{ix\omega}}dx$. Show ...
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41 views

Convergence of Fourier series in $L^2$ space

I'm learning about Fourier series, specifically $L^1$ and $L^2$ convergence, and need help with the following exercise: True or False (justify): $(1)$ The trigonometric series $2 + \sum_{k = ...
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1answer
59 views

When do Fourier series converge uniformly almost everywhere?

Let $f:\Bbb{R}\to\Bbb{R}$ be $2\pi$-periodic, $f\in L^1[0,2\pi]$, with fourier series $\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n \cos n\theta + b_n \sin n\theta)$. What condition must we impose upon $f$, ...
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1answer
30 views

Convolution with Gaussian question

Let $K_\epsilon(x):=\dfrac{e^{-x^2/\epsilon^2}}{\epsilon\sqrt\pi}$ for $x\in \mathbb{R}$, $\epsilon>0$. Let $f\in L^\infty(\mathbb{R})$ where $f$ is of bounded variation on any interval $[a,b]$. ...
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73 views

Problem: If $f\in L^1[0,1]$, show that $\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(x+\frac{k}{n})=\int_0^1 f(s)ds$ in $L^1[0,1]$.

This is the solution from the Prof.: Define $f_n(x)=\frac{1}{n}\sum_{k=0}^{n-1}f(x+\frac{k}{n})$. Then, unless $j<n$, we get $\hat f_n(j)=\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi i j k/n} \hat f(j)=0$ ...
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1answer
21 views

What's the meaning of the continuity in spectrum analysis?

We all know for any kind of signal $f(x)$, it corresponds a Fourier transform function $F(\omega)$ that $$ f(t) = \frac{1}{2\omega} \int_{-\infty}^{+\infty} F(\omega) e^{i\omega t} d\omega $$ So, $F(\...
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Help needed in deriving the complex root of raised cosine power spectrum

The raised cosine power spectrum (energy spectrum) is defined as $$|\hat{\phi}|^2 = \begin{cases} 1,& \text {if} |\omega| \leq \pi (1-b)\\ \dfrac{1}{2}[1+\cos\dfrac{|\omega|-\pi(1-b)}{2b}],& \...
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2answers
91 views

If $\int_0^{2\pi} q = 0$, then $\lim_{n \to \infty} \int_0^{2\pi}p(x)q(nx) = 0$

I'm learning about Fourier series and need help with the following exercise: Let the functions $p, q \in L^1([0, 2\pi])$ be bounded and $2\pi$-periodic. If $\int_0^{2\pi} q = 0$, show that $...
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26 views

Fourier Transform of $\frac{sin^2(3\pi C t) }{ 3 \pi^2 C t^2 }$?

C is a positive constant. How would you calculate the Fourier Transform of $\frac{sin^2(3\pi C t) }{ 3 \pi^2 C t^2 }$? As it is not easy to calculate the Fourier integral $ \int_{-\infty}^{\infty} \...
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9 views

When is a linear combination of spherical harmonics positive?

Let's say I have a real function $f$ on the sphere, and I express it as a linear combination of (real) spherical harmonics: $$f(x) = \sum_{l,m} \alpha_{l,m} Y_{l,m}(x).$$ Are there necessary and ...
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2answers
107 views

What is the Fourier Sine Transform of $\frac{1}{x(a^2+x^2)}$? (Please check my solution)

My textbook says that the answer is $\frac{\pi}{2a^2}(1-e^{-as})$ while my answer is $\frac{1}{a^2}\sqrt{\frac{\pi}{2}}(1-\frac{e^{-as}}{2}+\frac{e^{as}}{2})$. Here is my solution: \begin{align} \hat{...
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25 views

Find the function whose Fourier sine transform is $\frac{s}{1+s^2}$? (Please check my solution)

There needs to a $\sqrt{\frac{2}{\pi}}$ before that integral, which I have left by mistake. I have used this result : $$\int_{0}^{\infty}\frac{\sin{a x}}{x}dx = \frac{\pi}{2}$$ where $a > 0$. My ...
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33 views

Finding the maximum of $|\widehat{f''}|$ for $f$ in terms of the Gaussian

Let $$f = \begin{cases} e^{-x^2/2} - e^{-2 x^2} &\text{if $x\geq 0$,}\\ 0 &\text{if $x<0$.}\end{cases}$$ I would like to find out $|\widehat{f''}|_\infty$. A good numerical bound -- of ...
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1answer
48 views

Fourier transform of log(t)*exp(-t^2)

Let $f:\mathbb{R}\to \mathbb{C}$ be the function defined by $$f(t) = \begin{cases} \log(x) \exp(-x^2) &\text{if $x\geq 0$}\\ 0 &\text{if $x<0$.} \end{cases}$$ What is the Fourier transform ...
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32 views

mapping monotonicity and limit properties of $f$ to Fourier transform $\hat f$

Question Let $\hat f$ denote the Fourier transform of $f$ (non-unitary, angular frequency). I'm interested in a class of $\hat f$ that is characterized by the differential equation $$\hat f ' (t) = - ...
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1answer
24 views

Fourier transform symmetry property proof

I am trying to write my own proof that: If $f(t)$ is odd then so is $F(\nu).$ Where $F$ denotes the Fourier transform of $f.$ What I want to show is that if $f(t)=-f(-t),$ then $F(\nu)=-F(-\nu).$...
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1answer
32 views

A proof of the fact that the Fourier transform is not surjective

Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$ 1) Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$. 2) Show that $f_n \star f_1$ is the Fourier transform of $g_n = \...
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1answer
27 views

In a proof of the Fourier inversion theorem

The following is the Fourier inversion theorem on Schwartz space and the beginning of its proof in Hunter's Applied Analysis (p.305): Would anybody elaborate how the one dimensional case could ...
6
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1answer
46 views

Fourier Transform of $\frac{1}{\sqrt{|x|}}$

I want to find the fourier transform of $\frac{1}{\sqrt{|x|}}$. I checked the table of common fourier transforms in Wikipedia, and I know the answer should be $$\sqrt{\frac{2\pi}{|\omega|}}$$ What I ...
6
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74 views

Rigorous justification for “complex” change of variable in integration

Suppose that I have $X_1,\ldots,X_n$ i.i.d. $\sim $ $X$ and $Y_1,\ldots,Y_n$ i.i.d. $\sim$ $Y$ for some continuous $X$ and $Y$. Consider the r.v.'s $\bar{X}=\frac{1}{n}\sum_jX_j$ and $\bar{Y}=\frac{1}{...
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82 views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?
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58 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
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55 views

Eigenfunctions of non-uniform convolution

Consider a non-uniform ("generalized"?) convolution operator: $$ A_h[f](t) = \int f(x)h(x,t)dx $$ I would like determine the eigenfunctions. In the "stationary" case where $h(x,t) = h(x-t)$ we have ...
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56 views

2D Fourier transform of characteristic function of stripe on xy plane

Given a stripe $X$ on the xy-plane, namely $X\subset\mathbb{R}^2$, with $X=\{(x,y)\,|\; mx-\frac{1}{2}t \le y \le mx + \frac{1}{2}t$} and its "characteristic" function $$ f(x,y) = \begin{cases} 1, ...
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15 views

Wave Equation: Distribution which maximises Entropy

Given the wave equation: $\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)f(t,x)=0$ and expanding $f(t,x)$ through a Fourier Transform: $f(t,x)=\int d\omega dk F(\omega,k)e^...
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1answer
21 views

What will be power spectral density?

If the autocorrelation of a random process is given as $$r_x(k) = \begin{cases} 10 - \vert k \vert &\mbox{if } |k| < 10 \\ 0& \mbox{else.} \end{cases} $$ What will be the DTFT of this ...
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1answer
25 views

Example of fourier series diverges at least two points

I know continuous function $f$ such that fourier series of $f$ diverges at one point. But, I don't know continuous function such that fourier series of $f$ diverges at least two point.
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50 views

From FT deduce the value of $\int_0^{\infty} {\sin^4 x \over x^4} dx$

We have the function $$f\left(x\right)=\begin{cases} 1+x, & -1\leq x\leq0\\ 1-x, & 0\leq x\leq1\\ 0, & \textrm{otherwise} \end{cases} $$ we computed its Fourier transform: $$\hat f(t) = {...
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51 views

Discrete Fourier transform of $e^{i \alpha n} \Theta(n)$?

I am trying to understand the discrete Fourier transform $\mathcal{F}[s_n](\omega)=\sum_{-\infty}^{\infty} s_n e^{-in\omega} $ of $$s_n = e^{i \alpha n} \Theta(n),$$ with the Heaviside step function $\...
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42 views

Convolution and Fourier transform problem

I was struggling with this question, can use some help. given that $a\not=0$ $$f_a(x) =\frac{1}{x^2+a^2}$$ I'm trying to find k and c dependent on a and b $$(f_a ∗ f_b) (x) = kf_c(x) $$ I know ...
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1answer
30 views

The indefinite integral of an $L^1$ function has a convergent series of Fourier coefficients

Let $f\in L^1([0,2\pi])$ be a $2\pi$-periodic function with $\hat f(0)=0$ and $\hat f(\vert n\vert)=-\hat f(-\vert n\vert)\geq 0$. Define $F(t)=\int_0^t f(x)dx$. I know that F iscontinuous, $2\pi$-...
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1answer
72 views

What assumptions do we need for Fourier transform of derivative formula

Suppose $f: \mathbb R \to\mathbb C$ is differentiable and $f$ and $f'$ are in $L^1(\mathbb R)$. Do we need further assumptions to have the formula: $$\widehat{f'}(t) = (2\pi it)\hat f(t) $$ My ...
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12 views

Is it possible to conveniently express the probability density of a vector by its Fourier transform?

Let $x$ and $\hat{x}$ be a random vector and its Fourier transform, respectively. For any practical purpose we can assume it's a finite vector and the Fourier transform is given by the DFT, i.e. there ...
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1answer
28 views

Proof of Fourier Shift Theorem

The Fourier transform of $f(x)$ is, by definition, $$ \mathcal{F}[f(x)](u) = \int f(x)e^{-ixu}dx.$$ Now, the Fourier transform of $f(x-a)$ is, according to the Fourier shift theorem, $$ \mathcal{F}[f(...
2
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45 views

Why is the inverse fourier transform of $e^{-\omega^2+2i\omega}$ of the form $e^{(x-2)^2}$ and not of the form $e^{(x+2)^2}$?

So wolframalpha gives me this: wolfram's result for inverse fourier transform but I have this relation in mind: fourier transform rule I'm using and combining this with the shift theorem I would ...