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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DFT of a sequence while ignoring some of its elements

I sample a signal at a certain frequency for a finite amount of time to get a sequence $$(x_n)_{n=1}^N = (x_1, x_2, ... , x_N)$$ with the intention of analyzing its power spectral density by ...
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260 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=...
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1answer
30 views

Approximation estimates in Sobolev spaces

Let's consider a bounded domain $\Omega \subset \mathbb{R}^d$, $d =2,3$, and let $\varphi$ be in $H^1(\Omega) \cap W^{1,\infty}(\Omega)$. Is it there a smooth (at least $W^{2,4}(\Omega)\cap W^{1,\...
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1answer
22 views

Convergence of Sequence of Fourier Transforms

Let's say I have a sequence of functions $f_n\in L^1(\Bbb{R})$ such that $f_n\rightarrow f$ in $L^1(\Bbb{R})$, and $\hat{f}_n\in L^1(\Bbb{R})$ for all $n$ (where hat is the Fourier transform), and $...
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20 views

Fourier transforms of some homogeneous functions

In $2D$, what are the Fourier transforms (in the sense of distributions) of functions of the form $x_i/|x|^2, 1/|x|, x_i/|x|, x_i x_j /|x|^2$, where $i = 1,2$. They are homogeneous and integrable ...
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1answer
477 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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22 views

Show that there is $f\in \mathcal C^0(S^1)$ s.t. $\lim_{n\to \infty }\|S_nf-f\|_{L^\infty }\neq 0$.

I have to show that there is $f\in \mathcal C^0(S^1)$ s.t. $$\lim_{n\to \infty }\|S_nf-f\|_{L^\infty }\neq 0.$$ The proof goes as follow : we know that $\|D_n\|_{L^1}\geq c\log(n)$ where $D_n$ is the ...
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505 views

Computing Fast Fourier Transform in R

This is a math question with programming application. So I am trying to find 3 things given a certain function in the x domain when transformed into the spectral domain. 1) the Amplitude 2) The ...
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22 views

How can we show that $\int_{|\alpha |\leq N}\hat f(\alpha )e^{2i\pi x\alpha }d\alpha $ converges to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$.

I am a Ph.D. student in statistic, and I wanted to know if there is an easy way to show that the Fourier inversion converge to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$. In other word, that $$\lim_{N\...
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60 views

Is a function $f:\mathbb R/\mathbb Z\to \mathbb R$ bounded ?

Let $f:\mathbb R/\mathbb Z\to \mathbb R$ a function (1-periodic). Is such a function bounded ? (it's the fact that f is defined on the circle that disturb me). Indeed, for such a function (usually at ...
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2answers
110 views

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

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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182 views

Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$

Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \...
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61 views

Definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis

The following is the definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis: Here a regular distribution is a tempered distribution $T_f$ such that it is given by $$ T_f(\varphi)=\...
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18 views

Complex conjugation of Fourier transform in frequency domain

If I have some function (which is an array in Python or Matlab) $F(\nu)$ and want to derive $F^*(-\nu)$, I can do an inverse Fourier transform on $F(\nu)$ to get $f(t)$, then complex conjugation on $...
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25 views

Integral, absolute value

Why did we get this identity? $$\frac{1}{\pi} \int_{0}^{2\pi} |x-\pi| dx = \frac{2}{\pi}\int_{0}^{\pi} \pi-x dx$$ And why do we integrate it to: $$-\frac{(\pi-x)^2}{2}$$ Why didn't we just ...
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27 views

How to calculate the integral of a fourier trasfom

I have to calculate this integral : $\int_{-\infty}^{+\infty} \hat G(\omega)e^{i\frac{\pi}{2}\omega}d\omega$ ($\hat G(\omega)$ is the Fourier trasform) with: $G:x\in \Bbb{R}\to \begin{cases} g(x), ...
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32 views

Pointwise convergence of Fourier series in two dimensions

By Carleson's Theorem, we know that for every $f\in L^2(\mathbb{T})$ $$ f(x)=\lim_{N\rightarrow\infty}\sum_{k=-N}^N\hat{f}(k)e^{2\pi ikx}\;\text{ a.e.} $$ Suppose now that $f\in L^2(\mathbb{T}^2)$. ...
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1answer
28 views

The Fourier transform of the Bartlett (triangular) window

I am trying to understand how to obtain the Fourier transform of the Bartlett (triangular) window. The Bartlett window is defined as $$ w_B(k)=\begin{cases}\frac{N-|k|}N,& |k|\le N;\\0,&|k|>...
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Is there a theorem that tells us that $\widehat{f\circ g}=\hat f(\hat g)$? [duplicate]

Is there a theorem that tells us that $\widehat{f\circ g}=\hat f(\hat g)?$ Here, $\hat f$ denotes the Fourier transform of $f$.
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Why $(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi}$?

Why $$(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi} ?$$ Indeed, to prove that the fourier transform is in the Schwarz ...
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134 views

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

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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1answer
45 views

How to calculate the Fourier trasform

I have to prove to what space $L ( \Bbb{R} )$ does not belong to the Fourier transform of : $G:x\in \Bbb{R}\to \begin{cases} g(x), & \text{if |x| $\leq$ $\pi$} \\ 0, & \text{if |x| $\gt$$\pi$...
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3answers
210 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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1answer
53 views

Significance of orthonormal basis in wavelet analysis

I've recently been looking into wavelet analysis and I have the question: What is the importance of wavelets having an orthogonal basis, say as opposed to a bi-orthogonal basis or otherwise? I'm ...
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34 views

How to derive the Fourier Transform of the cosine function?

Given that the function $f(t) = A\cos(\omega t-\phi)$. I cannot get the results for the $f$ domain transform $F(f)$ and the $\omega$ domain transform $F(\omega)$ to be equivalent
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Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem

This question spanned from a previous interesting one. Let $k$ be a real number greater than $2$ and $$\varphi_k(\xi) = \int_{0}^{+\infty}\cos(\xi x) e^{-x^k}\,dx $$ the Fourier cosine transform of a ...
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1answer
30 views

Calculating integral value of Fourier series

Given fourier series: $$\mathrm{S}\left(x\right) = {3 \over \pi}\sum_{n = 0}^{\infty} {\sin\left(\left[2n + 1\right]x\right) \over 2n + 1}\,,\qquad \left\langle -\pi,\pi\right\rangle $$ Evaluate: ...
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1answer
57 views

How is the study of wavelets not just a special case of Fourier analysis?

As far as I can tell, "wavelets" is just a neologism for certain "non-smooth" families of functions which constitute orthonormal bases/families for $L^2[0,1]$. How is wavelet analysis anything new ...
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1answer
45 views

Is Fourier series $L^2$?

Let $f\in L^2(0,1)$. I was wondering if the Fourier series of $f$ is a linear map $L^2(0,1)\to L^2(0,1)$. The linearity is obvious, but if $f\in L^2(0,1)$ does $S(f)\in L^2$ or not ? I tried as follow,...
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41 views

Calculate $\int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt$

Let $f\in L^2$. Knowing that $f$ has a Fourier transform given by $\hat{f}(w)=\frac{w}{1+w^4}$ calculate: $$\int_{-\infty}^{\infty}\frac{\sin(t)}{t}f(t)dt$$ Im having some trouble in trying to solve ...
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8 views

DFT of a square signal

$f_n = 1, n = N/4, ... , 3N/4 -1$ and $0$ otherwise (the signal being N-periodic). I'm trying to arrive at a simplification for $F_k$. $$F_{2k} = \frac{1}{N}\sum_{n=N/4}^{3N/4-1}W_N^{-2kn} = \frac{1}...
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1answer
23 views

Fourier Series of this function

Find the Fourier series of this function, only by using sine functions. This is not a homework, I'm just practicing different problems for an exam. I know that all coefficients, except b0 should be 0. ...
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38 views

Additive combinatorics modulo $N$: Reference request

For integers $N, t \geq 1$, would you know of any special sets $A$ of integers in literature for which either an explicit formula (hopefully nice enough) or good estimate is known for the number $$ \#\...
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1answer
37 views

When is a Fourier transform of a function continuous

I have read about a following claim in a book but cannot prove it. If a function $\psi(x)$ fulfills $\int dx|\psi(x)|^2|x|^2<\infty$, its Fourier transform $\phi(k)$ is continuous and nearly ...
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21 views

Fourier Analysis for Derandomization of Functions

I was wondering if there was an extension to Fourier Analysis on Boolean Functions. Specifically, it's well known that for any boolean function $$f: \{-1,1\}^{n} \rightarrow [-1,1] $$ we can decompose ...
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1answer
143 views

Expand $f(x) = x$ in a cosines serie

The task is: Expand $f(x) = x$ in a cosines serie. My approach was to use the trigonometry fourier formula $$ c_0 + \sum_{k=1}^{\infty } a_k \cos (k \Omega t) + b_k \sin (k \Omega t) $$ set $b_k = ...
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: $$\Lambda(n)=\...
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1answer
75 views

Finding $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$

I heard there were functions $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$. Is there a concrete example of such functions ? Thanks in advance !
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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
50 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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13 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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1answer
13 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
20 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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33 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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1answer
26 views

Sequence $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$ and Paley-Wiener space $PW(0,1)$.

Let us consider the Paley-Wiener space: $$PW(0,1):=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset (0,1) \}.$$ Let us consider $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$, for $x\...
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1answer
58 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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45 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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25 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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14 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. $f$ ...