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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2
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1answer
13 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 ...
1
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2answers
54 views

Coefficient calculation on Fourier series !? [on hold]

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}+ ...
0
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1answer
9 views

Is it possible to reconstruct signal using phase/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 is it possible to reconstruct given ...
0
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1answer
89 views

Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
2
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0answers
27 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 ...
1
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0answers
25 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 $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
32
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3answers
950 views

Instructive proofs in functional analysis

I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
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0answers
13 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 ...
0
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0answers
41 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 ...
3
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0answers
30 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 ...
3
votes
3answers
1k views

Sum of Sinusoids with Same Frequency = Sinusoid (proof)

I am studying Fourier analysis on my own, I realised that probably the first thing you want to proof in Fourier transform is that the sum of 2 sinuoids (namely a sine and cosine) with the same ...
0
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3answers
147 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
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0answers
7 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. Could someone please ...
3
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1answer
34 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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0answers
15 views

A linear response system with a periodic input

I'm currently trying to solve the following exercise: A linear system is driven by a periodic input $f(t)$ such that $f(t+T)=f(t)$. The response $g(t)$ of the system is such that a sinusoidal ...
2
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1answer
54 views
+50

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 ...
0
votes
1answer
16 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 ...
4
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1answer
289 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
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39 views
+50

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 $n\geq 1$ : $$|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi kx)}{k}\right| \leq C -\log |\sin (\pi x)|, \quad \forall x\in (0,1]. $$ ...
1
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1answer
284 views

Discrete Fourier Transform: Understand Negative Frequencies

I am trying to learn DFT on my own. I have been struggling for a while now around understanding the concept of negative frequencies and notably what happens when $k$ is greater than $N/2$ in the ...
0
votes
1answer
19 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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0answers
10 views

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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0answers
14 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 ...
286
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25answers
28k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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0answers
12 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 ...
0
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0answers
37 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 ...
4
votes
1answer
60 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}< ...
1
vote
2answers
278 views

Bijectiveness, injectiveness and surjectiveness of Fourier transformation defined on $L^p(\mathbb{R}^n), p\in [1, 2]$

For Fourier transformation defined from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$, $p\in [1, 2]$, and $1/p+1/q=1$, I heard that when $p=2$, FT is bijective. Is $p=2$ iff FT is bijective, FT is ...
1
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1answer
38 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 ...
0
votes
1answer
35 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 ...
2
votes
3answers
315 views

Morlet's wavelet reconstruction formula

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$ In order to reconstruct the ...
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0answers
41 views

wavepacket in one-dimensional quantum mechanics [on hold]

the problem is in the picture. The problem is about fourier series, but I do not see how it is related.
0
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1answer
24 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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0answers
25 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) . ...
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0answers
21 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 ...
0
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0answers
17 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 ...
0
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0answers
24 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 ...
2
votes
1answer
25 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 ...
1
vote
0answers
14 views

What is Phase congruency?

I am beginer in signal processing. I am studying the importance of phase in signal.I want to know about Phase Congruency but there is very little information available on internet . So can anybody ...
2
votes
1answer
438 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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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: ...
0
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1answer
341 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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0answers
22 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 ...
1
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1answer
41 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
vote
1answer
53 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 ...
4
votes
1answer
73 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 ...
1
vote
1answer
48 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
0
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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 ...
0
votes
1answer
26 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
42 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 ...