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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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A comparison of fourier series and fourier transform

I have been studying fourier series and fourier transforms and I'm confused about a lot of things mentioned in various questions etc. As an example, I'm a quite puzzled on why fourier transform is ...
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Fourier odd and even extension of descending steps function

Can anybody help me find even and odd extension of the descending steps function if $f(x)$ be: \begin{cases} f(x)= 1 & x\in[0,1]\\ f(x)= (0.5) & x\in[1,2] \end{cases}
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What is Fourier transform of $|(x,t)|^{-\alpha}$?

Let the $x\in \mathbb{R}^d, t\in \mathbb{R}$, i.e. $(x,t)\in {\mathbb{R^{d+1}}}$. I already know the Fourier transform of $|x|^{-\alpha}$ is $|\xi|^{-d+\alpha}$. How do I get the Fourier transform ...
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Convergence of $L^2$ functions with conditions on their Fourier transforms

Let $\{f_n\}_{n\in\mathbb{N}}$ be a Cauchy sequence of $L^2(\mathbb{T})$ functions, where $\mathbb{T}=[-\pi,\pi)$. Assume that, $\forall n\in\mathbb{N}$ and for $k\geq 0$, the following inequality ...
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Support of a fundamental solution of wave equation

I need to prove that the following fundamental solution of the wave equation: $$ E(x,t)= \mathfrak{F}^{-1}\Bigg(\frac{\sin ( t|.|)}{|.|}\Bigg)\frac{\theta(t)}{(2\pi)^{n/2}}(x) \in \mathcal{S}'(\mathbb{...
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Help with explanation to the notations in a paper about notations and use of Fourier and LaPlace Transformations

I'm looking at a paper 'Recent applications of fractional calculus to science and engineering' (https://www.hindawi.com/journals/ijmms/2003/753601/abs/) but some of the notations in it baffled me and ...
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How to sketch a graphs for Fourier series coefficients [on hold]

I'm struggling to plot the correct Fourier series graph for the following function when $n = 1, n = 1.5$ and $n = 2.5.$ $$y = 5 + 20 \sum_{n = 1}^{\infty} \frac{1 - \cos(n \pi)}{\pi^2 n^2}\cdot{\cos(...
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Laplace transform and frames vs Bases

The Laplace transform $$F(s) = \int^{∞}_{0}f(t)e^{-st} dt$$ can be understood much like the fourier transform, as a change of basis of an $L^2$ function to the eigen functions of the differential ...
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How do I induce this Hardy-Littlewood-Sobolev inequality

The Hardy-Littlewood-Sobolev inequality in my book, Let $0<\alpha < d$, $1<p<q<\infty$, and $\frac{1}{q} +1 = \frac {1}{p} + \frac {\alpha}{d}$. Then $$ \lVert f \ast |\cdot|^{\...
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extension of a regular path

Given a smooth path in $\gamma: I \to \mathbb R^n$. such that $\gamma(0) = x$, $\gamma(1)=y$, $\gamma'(t) \neq 0, \forall t$, $x \neq y$. Let $z \neq x, z \neq y$, Is it always possible to extend $\...
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Determining stochastic process to give a specified autocorrelation function

I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, \text{ and } R_Y[k] = u, \mid k \mid > 1$$ Part A Specify a choice of $u$, ...
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Properties of laplace type transform of $t^{\alpha - 1}$

Let $p>2, \frac{1}{p} < \alpha < 1- \frac{1}{p}$ and define $g_\alpha(t) := t^{\alpha - 1} \chi_{[1, \infty]}$. Then $g_\alpha \in L^p(\mathbb R)$. Define $$f(z) := \int_1^\infty g_\alpha(t) \...
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Fourier series using other orthogonal systems

By definition, a set of functions $S = \{\phi_1,\phi_2,...\}$ is called an orthogonal system on the interval $[a,b]$ if $\forall\phi_i,\phi_j,i\neq g: (\phi_i,\phi_j)=0$. Examples: 1) $S_1 = \{1, \...
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How to approximate identity function using Fourier sine series

I want to approximate identity function $g(x) = x$ for $x \in [0,x_c]$ with $x_c<\pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x \in [0,\pi]$, $f(x)$ is assumed ...
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What is meant by “symplectic Fourier transform”?

I've recently come across the term symplectic Fourier transform (see this paper, first page, second column), but googling didn't lead me to any satisfactory explanation of what is meant with this term....
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Difficult Fourier integral giving a distribution

I would like to understand the distribution defined by $$ b(x)=\int_{-\infty}^{\infty}\lvert y\rvert e^{-ixy} dy $$ What I've understood so far is that $$ b(x)=\lim_{\alpha\to0^+}\int_{-\infty}^{\...
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Fitting a sinusoid vs. DTFT

I want to fit a sinusoid to a given discrete finite real signal $X(n)$ with length $N$. Of particular interest is the frequency. For the estimation for example least squares can be utilized: $$\text{...
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1answer
49 views

Real positive index Sobolev spaces are Hilbert spaces

I'm trying to prove that, for $k\geq 0$, Sobolev spaces defined in this way: $H^k(\mathbb{T})=\{f\in L^2(\mathbb{T}): \sum_{n=-\infty}^{+\infty}(1+n^2)^k|\hat{f}(n)|^2 < +\infty\}$ are Hilbert ...
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Forward and Backward Projections

I have the transform functions (forward and backward projections) such as: $$FP\{f(x,y)\} = \int_{-\infty}^{\infty}f(r\cos(\theta) - z\sin(\theta), r\sin(\theta) + z\cos(\theta))dz$$ $$BP\{g_{\theta}(...
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1answer
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Working partial fractions to solve for inverse Fourier Transfrom

I have a system where $H(w) = \dfrac{1}{(1-\frac{1}{4}e^{-jw})(1-\frac{1}{3}e^{-jw})}$. I need its inverse discrete Fourier transform. My thinking is that I could use partial fraction decomposition ...
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Interchanging limit and integral.

Suppose $(X,\mu)$ is a probability space, $W\in L^1(X)$, $V\in L^\infty(X)$, and $V_n\to V$ in $L^2(X)$ (in my situation $V_n$ is the partial Fourier sum and so the $L^2(X)$ convergence is automatic). ...
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Series of $\sin(nx)$ terms that sum up to $0$.

Working in $\mathbb{R}$, what sequences $a_n$ satisfy $\sum_{n=0}^{\infty} a_n \sin(nx)=0$ for all $x$, pointwise ? I've never thought about this and I'm not sure whether I'm not missing something ...
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DTFT of odd function?

I am confused about the result of an odd function being purely imaginary for the DTFT of an odd function. I was under the understanding that the result of a FT was amplitude and phase as a function of ...
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1answer
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Use Fourier coefficients of $f(t)=t$ to show $\sum_{n=1}^{\infty} \frac{1}{n^2}= \frac{\pi^2}{6}$ [duplicate]

Using Fourier coefficients of $f \in L_2(\mathbb{T})$ given by $f(t)=t$ for almost all $t \in \,\,]-\pi, \pi[$ show that $$\sum_{n=1}^{\infty} \frac{1}{n^2}= \frac{\pi^2}{6}$$ I think it’s an easy ...
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If $\int_{a}^{b} f(x) g(x) dx=0$ with $f(x)=\sum_{i=0}^{\infty} a_n x^n$, can I integrate term by term?

Suppose $\int_{a}^{b} f(x,l) g(x) dx=0...(1)$ with Taylor series of $f(x)=\sum_{i=0}^{\infty} a_n x^n$, where $a_n$ depends upon parameter $l$. I want to construct non-zero bounded integrable function ...
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Show that $g_1(\mu_1,\sigma_1)\ast g_2(\mu_2,\sigma_2)=g\left(\mu_1+\mu_2,\sqrt{\sigma_1^2+\sigma_2^2}\right)$

The definition (convention) I have been using for the Fourier transform is $$\mathscr{F}[f(t)]=g(\omega)=\frac{1}{\sqrt{2 \pi}}\int_{t=-\infty}^{\infty}f(t)e^{i\omega t}dt\tag{1}$$ and the inverse as ...
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How to find " The complex Fourier coefficients Xn

In the picture I have the question with my attempt , I could find everything without problem but the 3rd branch's asking about Xn (Fourier coefficients ) since I have X3 and X4 are equal 0 , so how ...
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Statement of Parseval's theorem for Fourier Transform

the following is the statement of Parseval's theorem from Wikipedia, Suppose that $A(x)$ and $B(x)$ are two square integrable (with respect to the Lebesgue measure), complex-valued functions on $\...
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Value of an integral depending on a function and its cosine transform

Let's consider $g(x)$ a function in $C^{\infty}(\mathbb{R}^+ \to \mathbb{C})$ such that $g(x)$ is asymptotic to $x^{\alpha}$ for $x$ near $+\infty$ with $g(0)=0$, and $\alpha$ a complex number such ...
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How to prove that $\left|\sum_{k=0}^{n-1}\sin(2k+1)x\right|$ is bounded

I'm currently studying Fourier series using a textbook which unfortunately provides only partial solutions to its explanations. To show the uniform convergence of a Fourier series I need to prove the ...
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Discrete Fourier transform of sampled sin function nothing like continuous?

So I'm a little confused about what is going on with the discrete Fourier transform. I tested out discrete Fourier transform with a little python script on a sin function ...
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The orthonormal basis $f_n(x) = e^{2\pi i nx}$ for $L^2(\mathbb R/\mathbb Z)$ defining Fourier series

I thought I understood this, but now I don't think I do. So the functions $f_n(x) = e^{2 \pi i n x}$ form an orthonormal basis for the complex Hilbert space $L^2(\mathbb R/\mathbb Z)$. This means ...
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1answer
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Fourier Expansion of a function on $\mathbb A_k/k$

Let $k$ be a number field, and let $\mathbb A_k$ be the ring adeles of $k$. The quotient group $\mathbb A_k/k$ is compact, and the choice of a nontrivial character $\psi$ of $\mathbb A_k/k$ gives an ...
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What is the Fourier transform of the 2 dimensional airy function?

What is the Fourier transform for the given two dimensional airy function, $$f(x,y) = \frac{J_1(r)}{r}\,.$$ Where $J_1$ is the Bessel function of the first kind, order one. And $r=\sqrt{x^2+y^2}$. ...
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2$\pi$-peridodic and continuous function with non summable Fourier coeffficients

Can someone tell me an example of 2$\pi$-periodic and continuous function, f, with Fourier coefficients $\hat{f}(n)\;\forall{n\in{\mathbb{Z}}}$ such that $\sum|\hat{f}(n)|>\infty$? Thanks.
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$f(x)=\sum_0^\infty\frac{\sin(2n +1) x}{2n+1}$ is the Fourier expansion of $(-1)^m\frac{\pi}{4}$ for $x\in (m\pi,(m+1)\pi),m\in Z$.

Suppose given $f(x)=\sum_0^\infty\frac{\sin(2n +1) x}{2n+1}$. I want to identify $f(x)$ as $(-1)^m\frac{\pi}{4}$ for $x\in (m\pi,(m+1)\pi),m\in Z$. I am only considering this function as $x\in R$ ...
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For a Schwartz function $f$, if $\int_{\mathbb{R}} f(x) x^n dx = 0$ for all nonnegative integers $n$, is $f$ identically 0?

This is an old exam question I'm practicing with. The associated hint is to use the Fourier transform. I'm pretty lost, but here are my thoughts so far. First, in this old stack exchange question a ...
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Fourier Transform of Exponentially Decaying Function Cannot Have Compact Support

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, with $|f(x)| \le e^{-|x|}$ a.e. Then how can we prove that its Fourier transform, $\hat{f}$, cannot have compact support (unless $...
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interpreting sound curves as sinogramms

What's the difference between a spectrogram--spectrum over time (${F}_f(t)$?)-- and a, whatsitcalled, soundwave plot--velocity over time $V(t)$--as often encountered in Music production? Are those in ...
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42 views

Calculating a periodic signal (way of solving this)?

I created my own examples so i can have the gist of how to solve the real ones that my homework needs so here we go: $$x(t)=\sum_{n=-\infty}^\infty \Pi\left({t-4n\over2}\right) + \sum_{n=-\infty}^\...
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Are topologies induced by following families of seminorms same?

Let $G$ be an open subset of $\mathbb R ^n$ and $D(G)$ denotes the set of smooth functions with compact support in $G$. Consider following families of seminorms, For $f\in D(G)$ $||f||_N = \sup \{...
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Function from weighted $L^p$ whith Fourier transform supported on finite measure set

Let $\mu$ be a positive ($\sigma$-finite) measure on $\mathbb{R}$ absolutely continuous w.r.t. Lebesgue measure. I am looking for a function $0\neq f\in L^p(\mathbb{R},\mu)$, for a fixed $p\in[1,\...
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Finding Fourier Series Trouble

There is a question in my homeworks and I couldn’t ask to professor since she has gone abroad. I don’t have any idea abot what should I do. I really need help. I have an exam this week. Any help will ...
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Finding the envelope frequency of a sinusoid (From a musical major triad) [closed]

[Editor 2’s introduction intended to address votes to close because the question wasn’t mathematical.] The trigonometric formula $\sin{(at)}+\sin{(bt)}=2\cos({a-b\over2}t)\sin({a+b\over2}t)$ can be ...
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If we use a sinusoidal signal as an input signal to a linear transmission path, then we always get out a sine wave of the same period/frequency

An Introduction to Information Theory: Symbols, Signals and Noise, by John R. Pierce, says the following: With the very surprising property of linearity in mind, let us return to the transmission ...
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How is Parseval’s theorem applied on this equation?

I am reading a paper and the paper mentioned the Parseval’s theorem. I googled a lot and I can't find how is the theorem apllied to the equation. The following is the confusing part of the paper. How ...
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54 views

When does “Gaussian integrability” imply regular integrability?

Let $\varphi:\mathbb R\to\mathbb C$, and suppose that the limit $$\lim_{\sigma\to\infty}\int_{-\infty}^\infty \varphi\left(x\right)\exp\left\{-\frac12\cdot\left(\frac x\sigma\right)^2\right\}\,dx$$ ...
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1answer
38 views

Trouble understanding steps in convolution

This problem comes from Brian Fisher and Biljana Jolevska-Tuneska's 2010 paper 'On the logarithmic integral'. As it is freely available I will refer to it rather than copy large chunks here that will ...
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54 views

Applying the residue theorem to calculate the Fourier transform of $\frac{1}{(x-\tau)^k}$

I'm trying to do this exercise from Daniel Bump's book, Automorphic Forms and Representations. For $f: \mathbb R \rightarrow \mathbb C$, the Fourier transform $\hat{f}$ is defined by $$\hat{f}(v) = \...
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37 views

Proving that real trigonometric polynomials are closed under multiplication

Recall that real trigonometric polynomials are functions of the form $$ f(\theta) = \frac{a_0}{2} + \sum_{k=1}^n (a_k \cos(k\theta) + b_k \sin(k\theta)). $$ I want to prove that real trigonometric ...