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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39 views

$\sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ does not converge as $\theta \rightarrow 0?$

We know that the series $H(\theta) := \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ is convergent for every $\theta \in (0,1)$ and for $\theta = 0$ the series tends to $+ \infty$. Is it ...
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1answer
36 views

Eigenvalues of Fourier Transform on Schwartz Functions

Find all the eigenvalues of the Fourier transform $\hat{f}$(viewed as an operator acting on the class of Schwartz functions $S(R)$), i.e. all values $\lambda \in \mathbb{C}$ such that there exists a ...
4
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1answer
114 views

Estimate $|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi \lambda_k x)}{\lambda_k}\right|$.

We known that there exists a constant $C\geq 0$ such that for all $n\geq 1$ : $$|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi kx)}{k}\right| \leq C -\log |\sin (\pi x)|, \quad \forall x\in (0,1]. $$ ...
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0answers
24 views

Parametrization of arbitrary objects to display on an x-y-scope

I am trying to find an approach for general parametrization of an arbitrary geometric object or closed curve. Though I am not sure if I am on the right path with that. Basically I have an geometric ...
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0answers
19 views

Range of Fourier Tranform 0n $L_1(\mathbb{R})$ is dense in $C_0(\mathbb{R})$

I want to prove it through the hint given in the notes available online(link provided below). It says first prove that if $f\in C_c^2(\mathbb{R})$, then $\hat{f}\in L_1(\mathbb{R})$; and hence ...
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0answers
28 views

Fourier transform of a 3sinc^2(100πt)

I'm currently studying for an exam, and I'm not sure the textbook's answer for the fourier transform of 3sinc^2(60πt) is correct. For this question, I incorporated the duality property. Below is my ...
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1answer
30 views

The proof of the Plancherel Theorem

I am reading the proof of the Plancherel Theorem in Folland. But I am quite confused about one of his claims. Suppose $f,g \in L^2(**T**)$ and $\hat{f}\in L^1$ ($\hat{f}$ is the Fourier transform). ...
2
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1answer
27 views

$k^2 e^{ikx} \rightharpoonup 0$ in the Sense of Distributions

So I'm concerned showing $k^2 e^{ikx} \rightharpoonup 0$ in the sense of distributions or in other words for any $\phi \in C_c^{\infty}(\mathbb{R})$ we have for any $\epsilon > 0$ $$ \left\lvert ...
3
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1answer
58 views

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...
5
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1answer
114 views

Improper integral Riemann sum limit in the derivation of Fourier series to Fourier transform

To give background to my question, in all the books I've looked at to derive the inverse Fourier transform of a continuous function $f$ on $\mathbb{R}$, they seem to work as follows. Let $k$ be a ...
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26 views

2D Fouier Series coefficent

I have a question. He gave this picture and square signal. Firstly he wanted me square signal fouier series then 1 3 harmonic.Then ı found it. The other question is wanted fourier series (2d) . ...
0
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0answers
23 views

Proving a fourier transform expression with green's formuls

Using Green's formula, show that: $${\cal F}\left[\frac{d^2f}{dx^2}\right]= -w^2F(w) + \frac{e^{iwx}}{2\pi}\left(\frac{df}{dx} - iwf\right) \\(evaluated\ from\ -\infty\ to\ \infty)$$ last part is ...
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1answer
55 views

Smoothness of inverse Fourier transform

Let $\hat{f}(\xi)$ be a smooth function on $\mathbb{R}^n$ that decays like $|D^\alpha_\xi \hat{f}(\xi)| \lesssim (1 + |\xi|^2)^{-\frac{1}{4}(1 + |\alpha|)}$, where $\alpha$ is a multi-index such that ...
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0answers
19 views

Norm convergence of approximations to the identity

Let $\varphi \in L^1(\mathbb{R}^d)$ be such that $\int_{\mathbb{R}^d} \varphi(x) \, dx = 1$. For each $\varepsilon>0$, let $\varphi_\varepsilon:= \varepsilon^{-d} \varphi\left( \dfrac x ...
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1answer
27 views

Bound on the sup norm for derivatives of a particular $C^\infty$ function

I'm reading textbook "A Primer of Real Analytic Functions" and on page 86 the following "obvious" claim is made: Let $|| \cdot ||$ be the sup norm on $[0, 2 \pi]$ and define function $f$ to be ...
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2answers
48 views

Fourier inverse of a function to get dirac

I'm trying to get the dirac function from a fourier inverse tranform: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw$$ It is this last step I am stuck on to get the conclusion. Original ...
1
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1answer
42 views

Proof of the discrete Fourier transform of a discrete convolution

Let the discrete Fourier transform be $$ \mathcal{F}_N\mathbf{a}=\hat{\mathbf{a}},\quad \hat{a}_m=\sum_{n=0}^{N-1}e^{-2\pi i m n/N}a_n $$ and let the discrete convolution be $$ ...
1
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1answer
44 views

Integral of $\int_{-\infty}^{+\infty}\left |{\frac{\sin{x}}{x(1+x^2)}}\right|^2\,dx $

So the first part of the questions asks us to find the Fourier Transform of $$ f(x) = \left\{ \begin{array}{ll} e^{y} & \quad {-\infty}<x < 0 \\ e^{-y} & ...
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1answer
43 views

Why the sum of the list is 4?

Wolfram Alpha says Sum[Sin(Pi*n/4)]/(Pi*n/4),{n,-Infinity,Infinity}] is equal to 4 but I don't know how to resolve it... In my signal and system homework,this ...
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0answers
24 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
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0answers
10 views

Errors of approximating continuous Fourier transform by discrete Fourier transform

In http://planetmath.org/approximatingfourierintegralswithdiscretefouriertransforms some error analysis of using DFT to approximate continuous Fourier transform is indeed done, but there are things I ...
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1answer
19 views

$h = \sum_{n=0}^\infty (ae^{j\omega})^n$ , how is the approximation of this equal to $\frac {1}{1-h}$

the question in the title. im working on a z- transform problem. to find the Z - transform of $x(n) = a^ncos(\omega n)u(n)$, u(n) being the step unit function essentially i come down to the answer ...
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27 views

find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
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1answer
23 views

Applying Fourier transform to a gaussian

Let $$G_\beta(w) = e^{\beta w^2}$$ Now I get the process of applying a fourier transform (or inverse) to get a new gaussian: $$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$ but in doing the ...
1
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1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
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2answers
42 views

Application of Plancherel/Parseval

Assuming $u,v\in L^1\cap L^2$, then how do you show that $$\int uv=\int \hat{u}\hat{v}$$ I tried using Plancherel, but didnt give any nice result. Any ideas/hints? Thanks
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10 views

Fourier transformation of Principal value distribution [duplicate]

I have the principal value distribution defined as $pv(\frac{1}{x})(\phi)=\int^\infty_0\frac{1}{x}(\phi(x)-\phi(-x))dx$ and I want to show that the fourier transform is given by $-\pi i\cdot ...
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0answers
25 views

Fast Fourier Transform of a function: WolframAlpha vs calculated result

I have the following function: $$\frac{3}{\sqrt{12}\cdot\cosh(x)}$$ I want to calculate the Fourier transform of this function. When calculated with WolframAlpha, I get as result: ...
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1answer
37 views

How to calculate convolution integral?

I know the formula for a convolution integral but how would you actually carry out one when you have two piece-wise defined functions? If you had $$ f(x) = \left\{ \begin{array}{ll} ...
2
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0answers
21 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
4
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2answers
54 views

Integral using Parseval's Theorem

How would I integrate $$\int_{-\infty}^{+\infty} \frac{\sin^{2}(x)}{x^{2}}\,dx$$ using Fourier Transform methods, i.e. using Parseval's Theorem ? How would I then use that to calculate: ...
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2answers
50 views

$\|(g\widehat{(f|f|^{2})})^{\vee}\|_{L^{2}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{2}}$ for some $r\geq 1$?

Let $g\in C_{c}^{\infty} (\mathbb R)$, and $f, |f|^{2}f\in L^{2}(\mathbb R)\cap C_{0}(\mathbb R)$ (where $C_{c}(\mathbb R)$ is the class of smooth functions with compact support and $C_{0}(\mathbb R)$ ...
4
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1answer
75 views

Estimating an integral using the Poisson summation formula

Consider a continuous $L^1$ function $f$ : $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ such that $supp$ $\widehat{f}$ $\subset$ $[-1,1]$ and $f(n)$ $\geq$ $0$ if $n$ $\in$ $\mathbb{Z}$. The problem is to ...
3
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3answers
34 views

How do I show $\int_{-\infty}^\infty \frac 1{(a^2+s^2)(b^2+s^2)} ds=\frac {\pi}{ab(a+b)}$ using the solution to the following Fourier transform?

For a function $f_a(x)=e^{-a|x|}$ , where $a>0$ I have found that the fourier transform of it is as follows, i know this is correct. $\def\F{\mathcal F}$ \begin{align*} \F(f_a)(s) &= ...
3
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0answers
38 views

Asymptotic expansion of a Fourier Transform as $\omega\rightarrow 0$

First of all, I do apologise if the question is not formulated in precise mathematical terms, but as a physics student I lack a formal background on rigorous functional analysis. Suppose we have a ...
0
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0answers
29 views

Exact formula for alias of Discrete Fourier transform for periodic sigals

Suppose that $f(t): \mathbb{R} \to \mathbb{C}$ is a $T$-periodic signal, with highest frequency $f_h$. Now suppose that our sampling rate frequency is lower than $f_h$, and is not any multiples of ...
1
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1answer
14 views

ODE with finite Fourier expansion periodic coefficients

Regard the ordinary differential equation $$ \dot a(t) = z(t) a(t) $$ where $a(t)$ and $z(t)$ are matrix valued such that $z$ is periodic ($z(t+2\pi)=z(t)$). Then it is well-known (Floquet theory), ...
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1answer
25 views

how do i find the fourier transform of the function $f_a(x)=e^{a|x|}$

Having trouble finding the fourier transform of the function $f_a(x)=e^{-a|x|}$ , where $a>0$ I currently have that $$ \mathcal{F}(f_ax) = \int_{0}^{\infty} e^{-ax}\, e^{ i s x} \,ds= ...
0
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1answer
35 views

Fourier transform, circular symmetry

I need to compute the two-dimensional Fourier transform of a function with circular symmetry: $$ \int dx dy\, \frac{e^{i (k_x x+k_y y)}}{((z'-z)^2-t^2+x^2+y^2)((z'+z)^2-t^2+x^2+y^2)} $$ For ...
1
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1answer
67 views

Lipschitz continuity of complex valued functions (Fourier basis)

Apologies if this is a rather stupid question, but as a student with no prior knowledge on complex analysis, I was wondering if the following function of $x$ \begin{equation} f(x)=e^{-iqx} ...
1
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1answer
46 views

Solving $\int_{-\infty}^\infty f(\tau) {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau$.

We know that, for any real numbers $\lambda$ and $\nu$, it has \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= ...
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1answer
31 views

Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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0answers
17 views

FFT of k*k matrix from FFT of a j*j matrix

FFT of matrix a j by j matrix, A $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ = $\begin{bmatrix}10 & -2\\-4 & ...
3
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1answer
39 views

$\|fg\|_{A (\mathbb T)} \leq C \|f\|_{L^{2}} \|g\|_{A (\mathbb T)}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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42 views

Represent 1/f(x) as integral

Suppose I have a function $f(x)=\int_a^b p(t)g(x,t)dt $ where $p(t), g(x,t)$ are known. I was wondering if there was a way of representing the reciprocal as an integral as well? i.e. ...
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0answers
17 views

Calderón–Zygmund lemma and Lebesgue measure

Can someone please explain the relationship between the Calderón–Zygmund lemma and Lebesgue measure, thanks.
2
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1answer
31 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
0
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1answer
46 views

Prove $\int {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= {\operatorname{sinc}}(\lambda-\nu ).$

I want to prove the following relation. For any real numbers $\lambda$ and $\nu$, we have \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) ...
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0answers
11 views

Fourier transform of spherical harmonics divided by $|\vec{x}|^{3}$

I need a formula for Fourier transform of spherical harmonics divided by third order polynomial of the form $$F^{m}_{l}(\vec{x}) = \frac{Y^{m}_{l}(\vec{x})}{|\vec{x}|^{3}}$$ Spherical harmonics are ...
1
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1answer
33 views

periodicity of an exponential sum

I wish to rigorously prove that the function $f(x), x \in \mathbb{R}$ is not periodic. A function is defined to be periodic with period $M$ if $f(x+M)=f(x), \forall x \in \mathbb{R}$. Here $f(x) ...