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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Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
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Is Fourier transform still writing a function as a series of sines and cosines?

In the Fourier series we write a function as a series of sines and cosines. Fourier transform seems to me to be totally different, we are not finding a series but rather a function $\hat f(w)$. So ...
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36 views

Fast Fourier Transform Splitting Algorithm

I'm trying to figure out how the FFT splitting algorithm works. I've pretty much understood the general idea, but when I try to compute it, I get something completely different than what I expect $ ...
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26 views

Convergence of $S_{n}(f;t)$

Let $f \in L^P(T)$ for some $p>1$. If $n$th Fourier partial sum $S_n(f;t)$ converges almost everywhere as $n \rightarrow \infty$, does the limit have to be $f(t)$ almost everywhere? I am trying to ...
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18 views

Fourier transform of Schwartz functions

I am stuck with the following question: Suppose $f \in \mathcal{S}(\mathbb{R})$ satisfies $\widehat{f}(\xi)=0$ for $|\xi|<1.$ Prove that there exists $g \in \mathcal{S}(\mathbb{R})$ such that ...
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Too strong assumption in the Uniqueness Theorem of Rudin's Real and Complex Analysis?

In Rudin's Real and Complex Analysis, there is the following result about Fourier transforms. The Uniqueness Theorem If $f\in L^1(\mathbb{R})$ and $\hat{f}(t)=0$ for all $t\in\mathbb{R}$, then ...
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50 views

Showing the range of Fourier transform on $L^1(\mathbb{R})$ is in $L^1(\mathbb{R})$ for a particular scenario

This is a problem from Katznelson's Harmonic Analysis book(Page 143, Problem 8): Suppose $f\in L^1(\mathbb{R})$ and is continuous at $0$. Also suppose $\hat{f}\geq 0$, then show that $\hat{f}$ is in ...
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What will happen if we try to reconstruct signal using phase only or magnitude only?

I am studying Fourier Transform and it's inverse. We get phase and magnitude from Fourier transform and reconstruct it back from both together My question is that What will happen if we try ...
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40 views

A question regarding Parseval's identity.

In most books/websites, Proposition 2 (see below) is either stated for a Hilbert space or proved via Riesz-Fischer. Does the follow approach (which seems to work in an inner product space) fall down ...
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35 views

Help for solving limi of the Complex Fourier Series

I need help for this exercise. Let: $ f:\left[ -T /2, T/2 \right]\rightarrow \mathbb{R}. $ I need show that $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
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Coefficient calculation on Fourier series !? [closed]

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
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44 views

How to compute the Fourier transform of $\operatorname{rect}$?

I am trying to compute the Fourier transform of $\operatorname{rect}$, where $$\operatorname{rect}(t) = \begin{cases}1 &, 0 \leqslant t \leqslant 1\\ 0 &, \text{otherwise.} \end{cases}$$ I ...
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11 views

phase offset and fourier transform

The English Wikipedia article on the Fourier Transform states that the complex argument of the Fourier transform is the phase offset of the basic sinusoid in that frequency. Initially I could not see ...
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24 views

Non Riemann summable Fourier series but Abel summable

A Riemann summable Fourier series is also Abel summable. I am looking for an example of a non-trivial Fourier series that is Abel summable at a point but NOT Riemann summable at the same point. Such ...
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42 views

How to compute $\int _\mathbb{R}\frac{sin^{2n}t}{t^{2n}}dt$?

If $n=1$ we can compute $\int _\mathbb{R} \frac{sin^{2}t}{t^{2}}dt$ by using Parseval's formula since $\widehat{1_{[-1,1]}}(x)=2\frac{\sin x}{x}$. We obtain $\int _\mathbb{R} ...
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20 views

Rectangular Width Fourier Function

Working on #7, I've tried writing out the Fourier transformation and plugging it into the formula and multiplying it with Wf, but I'm getting mixed up about how I'm allowed to combine integrals and ...
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Discrete fractional fourier transform matrix

I am trying to write a matlab code for some calculations based on Discrete fractional fourier transform. in this article: Optimal filtering in fractional Fourier. after equation (7) a notation Fa is ...
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19 views

Gradient of the Fourier transform of a function

I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function. In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is ...
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30 views

Deflection of String

I am trying to determine u(x,t) for a string of length L=1 and c^2=1 when the initial velocity is 0 and initial deflection with small k(.01) is as follows: ...
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19 views

How to construct a continuous function that is (mean) convergent to a given square integrable function

(In the Riemann Sense, this is a lemma before the Fouriers-Mean-Convergence Theorem) Suppose we have a square integrable function f:$[0,2\pi]\rightarrow \mathbf{C}$. We know that $\int_{a}^{b} f^2 ...
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42 views

Fourier transform of a tough composite function (sinc, sqrt, polynomial…)

Is it possible to compute the Fourier transform of $\mathrm{sinc}(\sqrt{1+x^4})$ in closed form? It appears the problem to be suited for contour integration, and I started to tackle the mere ...
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62 views

Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant. Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< ...
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$\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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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 ...
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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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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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20 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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30 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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32 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). ...
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$k^2 e^{ikx} \rightharpoonup 0$ 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 ...
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71 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 ...
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121 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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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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56 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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24 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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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 ...
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1answer
50 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 $$ ...
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1answer
46 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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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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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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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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$h = \sum_{n=0}^\infty (ae^{j\omega})^n$ , 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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1answer
25 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 ...
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12 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
47 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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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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28 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
40 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} ...
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24 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 ...