Questions tagged [fourier-analysis]

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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26 views

Fourier transform of product of exponential decay and cumulative normal

I am trying to find the Fourier transform $ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(t) \cdot e^{i\cdot\omega\cdot t} dt $ of the following function: $$ f(t) = e^{-a\cdot t}\cdot \mathcal{N}\...
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39 views

If $\{x\}:=x-\lfloor x\rfloor$, then are these the same? $\{l+m\}$, $\left\{l+\{m\}\right\}$, $\left\{\left\{l\right\}+\{m\}\right\}$

I am currently working on some signal processing project and come across this particular problem: Define $\{x\} := x - \lfloor x\rfloor$ and consider $$y_1 = \{l+m\}$$ $$y_2=\left\{l+\{m\}\...
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18 views

implication of $\int _\mathbb{R} \left(|\hat{f}(x)|^2+|\hat{f}(x)|^2/|x|\right)dx < \infty$

Let $f$ be in $L^1(\mathbb{R})$ and $\hat{f}$ be its fourier transform. Then, what can be the implication of the condition $$\int _\mathbb{R} \left(|\hat{f}(x)|^2+|\hat{f}(x)|^2/|x|\right)dx < \...
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32 views

Book recommendation on Fourier analysis technique for PDE

I want a graduate-level textbook which discusses Fourier analysis techniques for solving PDE. To elaborate what I want to study, consider the Laplace operator $-\Delta$. Since the differential ...
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23 views

Estimate of a certain Hardy-Littlewood maximal function at infinity

$f\in L^1_{loc}(\mathbb{R}^n)$,prove that if $f(x)=O(|x|^{-n}),|x| \rightarrow \infty$,then $Mf(x)=O(|x|^{-n}\log|x|),(|x|\rightarrow\infty)$. where $Mf$ is the Hardy-Littlewood maximal function of f. ...
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82 views

Why are variance and expected value all we care about?

My statistics background is almost $\varnothing$, so I apologise if my question has already been asked, and I just didn't know the right terminology to find it here. Suppose $p_X$ is the probability ...
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27 views

“Steady-state” of Error function using Fourier Transform

I am interested in studying the steady state dynamics of a function, by means of looking at the Fourier transform. As a way to illustrate, suppose that the signal I want to study is the error function....
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error estimation for the Fourier series of the fractional part of $x$

I'd like to prove the error estimate for the factional part of $x$ as given above. To begin with, we apply the Abel's formula and get $$(1):\left|\sum_{n=N+1}^M \frac{\sin(2\pi n x) }{n}\right|=\...
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10 views

Coefficient of variation of a periodic signal

Is there a way to estimate the coefficient of variation $\frac{\sigma}{\mu}$ of a non-zero mean periodic signal $x(t)$ from its Fourier Transform? I'd like to estimate the ``flatness'' of the signal
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33 views

Steady-state of a system and Fourier Transform

I successfully found the Fourier Transform of a function $f(t)$. Suppose the Fourier transform looks like the one below, so its not easy to calculate the Inverse transform: $$ F(w) = a ^{c - \sqrt{b -...
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Derivative of Fourier Expanded Variable

Suppose I have a vector field that is periodic in one direction such that it can be simplified as: $$\mathbf{u'}(x,y,z,t)=\int_{-\infty}^{\infty}\mathbf{\hat{u}}(x,y,t)e^{ikz} dk,$$ where $\mathbf{u'...
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How to visualize complex Dirichlet Kernel?

We know that the Dirichlet Kernel is defined as $$ D_{N}(t) = \frac{1}{2\pi} \sum_{n=-N}^{N} e^{int}$$ for $N \in \mathbb{Z}$ and $t \in \mathbb{R}$ (for Fourier series, we usually just consider the ...
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22 views

For every $f \in C(\mathbb{T})$, there exists a corresponding solution $u(x,t)$ to the heat equation on a ring under specific conditions

The following is an old exam question I'm stuck on: Show that for every $f \in C(\mathbb{T})$ (where $\mathbb{T} = [-\pi, \pi]$) there is an initial condition $g \in C(\mathbb{T}$ for which there ...
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44 views

Fourier transform of $\frac{1}{\sinh(x+a)}$ for complex $a$

I am trying to understand how to generally compute the Fourier transform of the function $\frac{1}{\sinh(x+a)}$, where $a$ is a general complex number. Plugging the equation into Wolfram Alpha gives a ...
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35 views

Fourier Expansion of this Wave.

I've been trying to Fourier expand $\psi(t)=\sin\left(2\pi at\ +\ \sin\left(2\pi bt\right)\right)$ skip the below paragraph if you like to I think the methods I've tried all along evolved with time. ...
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49 views

If $f \in L^1(\mathbb{R})$ and its Fourier transform has compact support, then $f(x) = \sum_{n \in \mathbb{Z}} f(n)\operatorname{sinc}(x-n)$

An old exam question I'm practicing with: Let $f \in L^1 (\mathbb{R})$ and assume that $\hat{f}$ (the Fourier transform of $f$) is supported on the interval $[-1/2, 1/2]$. Let $\operatorname{sinc} (x)...
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27 views

$2\pi$-periodic $L^2$ functions on $R^1$ approximated by its Fourier series

I'm reading section 4.26 in Big Rudin, but I have two questions. Suppose $f$ is in $L^1(T)$. This means $f$ is the class of all complex, $2\pi$-periodic, and Lebesgue measurable functions on $R^1$ for ...
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44 views

Fourier Series for a Dirac Train

I'm trying to find out by myself the Fourier Series of a Dirac Train, but I'm getting after Integration by Parts that Dn equals to 0 and not to 1 as needed to be. Could you please help me find my ...
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27 views

Fourier coefficients of $\frac {a-b \cos x}{a-be^{-ix}}H_1^{1} \left(\sqrt{a^2+b^2-2ab \cos x} \right)$ [closed]

The function is this $$\frac {a-b \cos x}{a-be^{-ix}}H_1^{1} \left(\sqrt{a^2+b^2-2ab \cos x} \right)$$ where H is a Hankel function. Is there any way to get the complex Fourier series coefficient of ...
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38 views

The fourier series of periodic and real analytic function

Let $f$ be a real analytic and periodic function defined on the interval $[0, 2\pi]$. Then $f$ is infinitely differentiable for sure. Therefore, the fourier coefficients of $f$ decay faster than any ...
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48 views

Proof of Theorem 9.10 in Rudin's Real & Complex Analysis

This theorem essentially states that an $L^p$ function can be approximated in the $L^p$ norm by convolutions with a suitable kernel, and I'm having a bit of trouble seeing how Rudin obtains one of the ...
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18 views

Wiener-Khinchin theorem problem

It is stated by the Wiener-Khinchin theorem that you can obtain the spectral density of a stochastic process $X_t$ merely by taking the Fourier-transform of its auto-correlation function \begin{...
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18 views

Multidimensional Fourier series

On $\mathbb{T}$, we have approximation series $\{f_n\}$ of $f\in L^p,p\in[1,\infty)$ in the form of $f_n(x)=\sum_{|k|\le n}a_{n,k}e^{ikx}$, converging to $f$ in $L^p$. (for example, we can consider ...
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41 views

Help to find complex Fourier series coefficient of this periodic function

I'm having big trouble finding the complex Fourier series coefficient of the following periodic function $$\frac{a-b\cos\varphi}{\sqrt{a^2+b^2-2ab\cos\varphi}}$$ Mathematica is unable to compute it!!...
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35 views

Impossible to find the complex Fourier series coefficient for this periodic function [duplicate]

I'm having big trouble finding the complex Fourier series coefficient of the following periodic function $$ f(\varphi) = \frac{a-b\cos{\varphi}}{\sqrt{a^2+b^2 - 2ab\cos{\varphi}}} $$ Mathematica is ...
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9 views

Find a certain function of moderate decrease

I am trying to solve the following problem: Find a function $f$ of moderate decrease such that $$\int_{-1}^{1} f(x-t) dt = e^{-2\pi |x-1|} - e^{-2\pi |x+1|} .$$ The first idea I had was to use the ...
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65 views

Distributional Fourier Transform of $\frac{1}{|x|^a}$

Let $h_a=\frac{\Gamma(\frac{a}{2})}{\pi^{\frac{a}{2}}}|x|^{-a}, x \in R^d$ Then $\hat{h_a}=h_{d-a}$ in the sence of $L^1+L^2-$Fourier transforms if $\frac{d}{2}<Re(a)<d,$ and in the sence ...
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21 views

Uniform convergence of fourier series of step function [closed]

Consider the series $f(x) = \sum_{n=1}^{\infty} a_n \sin nx$, where \begin{align} a_n = \begin{cases} \frac{4}{n \pi}, &n \text{ odd} \\ 0, &n \text{ even} \end{cases}. \end{align} It is ...
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14 views

Elliptic Fourier fit coefficients

I am trying implement a function that could fit an elliptic fourier curve on a set of border points of a detected object. I am using cv2.findContours to acquire border points from a binary image. Next ...
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1answer
57 views

Is there a Mellin transform or an analogue on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$?

From Wikipedia, the Mellin transform is an isometry $M : L^2(\mathbb{R}^+) \mapsto L^2(\mathbb{R})$, $$\{M f\} (s) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}^+} x^{-1/2 + \mathrm{i} s} f(x) dx.$$ ...
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1answer
29 views

Twice continuously differentiable wavelets with compact support

Do there exists wavelets such that both mother wavelet and the scaling function have compact support and they are both twice continuously differentiable?
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39 views

Fourier Series of Brownian Motion $(B_t)_{t \in [0,1]}$ wrong?

In our lecture about Brownian motion, we calculated the fourier series of Brownian motion $(B_t)_{t \in [0,1]}$. We did the following: We first defined the Brownian bridge $(\beta_t)_{t \in [0,1]}$ ...
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80 views

Carleson-Hunt Theorem on $\Bbb R$.

Wikipedia states the following: If $p\in(1,\infty)$ and $f\in L^p(\Bbb R)$ then $f(x)=\lim_{A\to\infty}\int_{|\xi|<A}\hat f(\xi)e^{i\xi x}\,d\xi$ almost everywhere. This seems clearly ...
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31 views

Tensor product of tempered distributions

If $u \in \mathscr S'(\mathbb{R}^n)$ and $v \in \mathscr S'(\mathbb{R}^m)$ are tempered distributions, then we can identify them with elements of $\mathscr D'(\mathbb{R}^n)$ and $\mathscr D'(\mathbb{R}...
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42 views

Calculation of Fourier Transform derivative $\frac{\mathrm{d}}{\mathrm{d}\omega}\ F\{x(t)\}(\omega)= \frac{\mathrm{d}}{\mathrm{d}\omega}(X(\omega))$

Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform, i.e. $\frac{\mathrm{d}}{\mathrm{d}\omega} F\{x(t)\}(\omega)$, in terms of regular Fourier ...
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23 views

Some criteria to determine series that could not be Fourier coefficients of some integrable function

Let $f\in L^1[-\pi,\pi]$ and assume that $\hat{f}(|n|)=-\hat{f}(-|n|)$. Then, it is wll-know that $\sum_{n>0} \frac{\hat{f}(n)}{n}$ is converget. Applying this fact, one may find some series (...
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22 views

Unstable behavior of Discrete Fourier Transform and Fourier Transform alternatives

I have come across a "weird" behavior of the Discrete Fourier Transform (which I will refer to from now as DFT). suppose $$x=(1,-1,1,-1)$$ The real part of the DFT of $x$ is $$Real(DFT(x))=(0,0,4,...
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30 views

Approximation of Fourier Transform by DFT

Let us say that we have some Schwartz function $f : \mathbb{R} \to \mathbb{C}$. The Fourier transform of $f$ is given by: $$ \hat{f} (\xi ) =\frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} f(x)e^{-2\pi i x \...
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38 views

Existence of inverse Fourier transform through validity of calculations

Let \begin{equation} f(x)=\left\{\begin{array}{cl}e^{-x}\,e^{-\gamma \min(x,k)}&,x\ge 0\\ 0&,x<0\end{array}\quad,\gamma>0.\right. \end{equation} $f$ clearly lies in $L^1$, so it's ...
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1answer
56 views

How to say Fourier series of continuous function is continuous or not?

I had found Fourier series of $f(x)=\cos(\frac{x}{2}) , -\pi<x<\pi.$ Its Fourier series is given by $\displaystyle S(x)=\frac{2}{\pi}\sum_{n=1}^{\infty}\left[\frac{(-1)^n}{n+0.5}+\frac{(-1)^{n+...
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137 views

A finite field problem

$\mathbb F$ is finite field, $\mathbb A$ is a subset of $\mathbb F$, and $|\mathbb A|>|\mathbb F|^{\frac 34} $. Proof $\forall x \in \mathbb F$, there exist $a,b,c,d,e,f\in\mathbb A$ that makes $x ...
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33 views

Fourier transform of a Weierstraß product

Consider $E_1:\mathbb{C}\rightarrow \mathbb{C}, s\mapsto(1-s)e^s$ and the infinite (and on $\mathbb{C}$ holomorphic) product: $$\Pi_k(s)=\prod_{q \in \mathbb{Z}^\times\setminus{k}} E_1\left(\frac sq\...
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1answer
23 views

Using Sobolev embedding for a high frequency tail

Suppose $f\in L^p(\mathbb{R}^n)$ for some $1\leq p<2$. Let $\chi$ be a smooth function which is 1 outside the ball of radius 2 and 0 inside the unit ball (we can take it to be radial if desired). I ...
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22 views

Pointwise convergence of Fourier-Legendre Series expansion

Recall the Fourier-Legendre expansion of any given function $f$ of $L^2(-1,1)$: $$\frac{1}{2}\sum\limits_{n=0}^{N}\left[(2n+1)\int_{-1}^{1}f(s)P_n(s)ds\right]P_n\underset{N\to\infty}{\...
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1answer
37 views

$\lim_{R\rightarrow +\infty}\int_{-1}^{1}\frac{\sin(2\pi R t)}{t}dt=\frac{1}{\pi}$

As suggested by Reuns in in this question. It is interesting that the problem there boils down to the following limit: $$\lim_{R\rightarrow +\infty}\int_{-1}^{1}\frac{\sin(2\pi R t)}{t}dt=\frac{1}{\...
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1answer
21 views

Any convex sequence induces a sequence $\{a_n\}$ to be the Fourier coefficients of a function $f\in L^1[0,2\pi]$

Let $\{a_n\}$ be an even sequence of non-negative numbers satisfying in $a_n\to 0$ and $a_{n-1}+a_{n+1}-2a_n\geq0$ for every $n>0$. We define $f=\sum_{n=1}^\infty n(a_{n-1}+a_{n+1}-2a_n)F_{n-1}$ ...
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2answers
105 views

Is $f(x)=\lim_{R\rightarrow \infty}\int_{-R}^{R}\hat{f}(w)e^{2i\pi xw}dw$?

Let $f$ be a function satisfying $$\int_{-\infty}^{\infty}|f(x)|dx<\infty.$$ Is it true that for almost every $x\in\mathbb{R}$, $$f(x)=\lim_{R\rightarrow+\infty}\int_{-R}^{R}\hat{f}(w)e^{2i\pi x w}...
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1answer
91 views

Proving a Mapping Property Between Tempered Distributions and Schwartz Functions

Say I have a Schwartz function $\varphi\in\mathcal{S}(\mathbb{R}^{2n})$. Consider a map $\psi(u)(x)=\langle u,\varphi(x,\cdot)\rangle$ for $u\in\mathcal{S}'(\mathbb{R}^{n}).$ I'd like to know how to ...
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43 views

Fourier transform in textbooks, $\mathcal{F}^{-1}(\mathcal{F})=I$? Lebesgue vs. P.V. integrals

In many textbooks, the Fourier transform of a $L^{1}(\mathbb{R}^{n})$ function $f$ is defined by $$\mathcal{F}(f)(w)=\int_{\mathbb{R}^{n}}f(x)e^{-2i\pi xw}dx.$$ Also, if $\mathcal{F}(f)$ belongs to $L^...
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1answer
26 views

Wavelets in Fourier domain

Given a wavelet family $\psi_{s, a}$ generated by translations and dilations of a mother wavelet $\psi$ $$ \psi_{s, a}(x)=\frac{1}{s} \psi\left(\frac{x-a}{s}\right) $$ we can show a wavelet ...