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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Extend a function 2pi periodically and calculate fourier

I have the function $$f(x)= \begin{cases} \frac{\pi}{2}+x & x \in (-\pi,0] \\ \frac{\pi}{2}-x & x \in (0,\pi]\\ \end{cases} $$ I need to extend it $2\pi$ periodically and then ...
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1answer
14 views

Decayment of Fourier coefficients of infinitely differentiable function

For a $C^n[-\pi,\pi]$ function $f$ we have that $|\hat{f}(k)|\in O(1/k^n)$. This implies that if $f$ is $C^\infty[-\pi,\pi]$ then its $k-th$ Fourier coefficient decays faster than any $1/k^n$, ...
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8 views

convolution of Schwartz functions with $f(x) = (1+\|x\|)^{-\frac{1}{2}}$

Let $f(x) = (1+\|x\|)^{-\frac{1}{2}}$ for $x \in \mathbb{R}^n$. This is clear that $f\star g \notin \mathcal S$ where $\mathcal S$ is algebra of Schwartz functions on $\mathbb{R}^n$ and $ g \in ...
3
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1answer
26 views

One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
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2answers
17 views

Fourier function expansion for extension over a $2\pi$ period

So I am currently looking at a fourier expansion for $$f(x)=\left\{\begin{array}{ccl}\sin x &\text{ if }& x\in[0,\pi]\\0 & \text{ if } & x\in[\pi,2\pi]\end{array}\right.$$ I am ...
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31 views
+150

Recurrence relation for Fourier-Legendre series.

So for the function $f(x) = \exp(-x)$ I have the formula for the coefficients of $$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$ which is(by using Rodrigues formula) $$a_n = \frac{2n+1}{2} ...
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1answer
37 views

Prove that $\mathscr{F}[f] \in L^2(\mathbb{R})$

Let $f \in L^2(\mathbb{R})$ (square integrable functions), I'm trying to prove that his Fourier transform also does: $\mathscr{F}[f] \in L^2(\mathbb{R})$. I have tried to bound it \begin{align} ...
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0answers
13 views

Show Fejer kernel on the real line is good, without using trignometric integrals.

This is from page 163 of Stein's Fourier Analysis. Fejer kernel on the real line is defined by $$ \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2$$ When $t=0$, ...
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18 views

Windowing effect and Fourier Transform

I understand how windowing effect helps to improve side lobes, of transformed signal in fourier spectrum. One way to explain this, is by pointing out that the sampled signal is considered periodic, ...
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0answers
17 views

Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)

On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula: $\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$ where $s>1$, $\chi$ is a ...
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1answer
29 views

Expansion theorem or Poisson Summation Formula? - Basis of eigenfunctions gives rise to a Fourier series

Does anyone could explain to me why in the Semiclassical's answer on the question Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula, the basis gives a series of the form ...
0
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2answers
32 views

Help with an Inverse Fourier transform

Can anybody please guide me how to compute the inverse Fourier Transform of: $$ f(k) = \frac{1}{1+k^2} \frac{\pi}{4}(\rm{sgn}(1-k) + \rm{sgn}(1+k)) $$
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31 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: ...
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0answers
20 views

$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
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3answers
28 views

A funtion and its fourier transformation cannot both be compactly supported unless f=0

Problem : Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$. Hint : Assume $f$ is supported in [0,1/2]. Expand $f$ in a ...
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1answer
22 views

If $f$ is continuous and moderate decreasing, then Fourier transform of $f$ is continuous.

If $f$ is continuous and of moderate decrease, show that $\hat{f}$ is continuous. My attempt: $$ \hat{f}(\omega+h)-\hat{f}(\omega) = \int_{-\infty}^\infty f(x)e^{-2\pi ix\omega}(e^{-2\pi ix h} - ...
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1answer
21 views

How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
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1answer
63 views
+100

Existence of operator

I want to show that for $ s> \frac{1}{2} $ there is a bounded linear operator $ T: H^s(\mathbb{R}^n) \to H^{s-\frac{1}{2}}(\mathbb{R}^{n-1})$ following the below steps: Consider that $ u \in ...
2
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1answer
1k views

How can I use the time-frequency uncertainty principle?

I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second. I have been told that, if I want to ...
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0answers
24 views

A function and its fourier transfrom cannot both be compactly supported unless f=0 [duplicate]

Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$. I assume that $\hat f$ is compactly supported function. Then, $\exists N$ ...
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1answer
17 views

Fourier part series, missing one piece

$$F(x)=\left\{ \begin{array}{rl} ax,&0<x<\pi,\\ bx,&-\pi<x<0, \end{array} \right.$$ So, far i've got: $$a_0 = - \frac{b\pi}{2} + \frac{a\pi}{2}$$ $$bn = \frac{1}{\pi} ...
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0answers
37 views

Fourier Tranform of $\frac {1}{(1+k \sin(t))^{3}}$

I'm stuck with some Fourier transforms that I'm not being able to solve and Mathematica is not helping: $\frac{1}{(1+k \sin(t))^{3}}$ (and the same one for sinh), where $k$ is a constant. Any ...
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0answers
13 views

Characterizing functions with controlled Fourier coefficiens

It's a well known fact that an infinite dimensional Banach space $E$ is not locally compact. One may consider, at which point, is this property lost, i.e. what kind of compact sets $K \subset E$ ...
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0answers
10 views

Control the value of a function at a point by the norm of its fourier transformation and itself

$n\leq 3$ ,$\Delta$ is the Laplacian on $L^{2}(R^{n})$, $Dom(\Delta) = \{\phi\in L^{2}(R^{n})|\Delta\phi\in L^{2}(R^{n})\}$. Please show that:for any $\phi\in Dom(\Delta)$,there exists constants ...
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0answers
9 views

What is the fastest way to show that FT Convolution theorem holds also in the case of a weighted sum?

Given $Y=X+Z$, with $X, Z $ r.v. such that $X \sim f(x)$ and $Z \sim g(x)$, the Fourier Transform Convolution property gets me the result: $$\mathcal{F}[(f \otimes g)(x)] = \hat{f}(\xi) \hat{g}(\xi) ...
0
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1answer
11 views

Why equals the z-Transform $c^n * u(-n-1)$? according to Matlab/WolframAlpha?

$x[n] = c^n * u[-n-1]$ Where u[n] is Heaviside step function. According to Matlab and WolframAlpha this equals 0. However if I compute the sum according to the z-Transform definition I got (sum from ...
3
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2answers
56 views

Fourier Transform Dirac Delta

I have recently learnt about tempered distributions, and how one can define the Fourier transform of a tempered distribution $v$ to be $\hat v$ so that $$\langle\hat v,\varphi\rangle=\langle v,\hat ...
3
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3answers
494 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
0
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1answer
36 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...
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0answers
29 views

Fourier transform without using Lebesgue measure

Let $\mathbb{L}^p(\mu)$ be a space such that $$ \mathbb{L}^p(\mu) = \left\{f:\mathbb{R}\to \mathbb{R} \mbox{ measurable}: \|f\|_{L^p(\mu)} = \left(\int_0^{+\infty} ...
2
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0answers
32 views

$f \in L_2$ bandlimited implies $f$ equal to continous function a.e. (without using Parley-Wiener)

I was wondering, if my proof is right as I didn't find any similar statements in books or the internet without using the Parley-Wiener-Theorem: If we have $f \in L_2(\mathbb{R})$, bandlimited (i.e. ...
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0answers
87 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
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0answers
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How to compute the ifft of a vector?

In the following post Concrete polynomial implementation it is said that the final step before obtaining the product of two polynomials is to compute the ifft of a vector. How to compute the ifft of ...
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1answer
28 views

Pointwise evaluation of $L_2$ Fourier Transform

We know, that the $L_2$-Fourier Transform of a function $f\in L_2$ is defined as a limit of $L_2$ functions (e.g. $\ \mathcal{F} f=\lim_{n\to \infty} \int_{-n}^{n} f\cdot \chi_{(-n,n)}\ d\lambda ...
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4answers
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What is the relationship between Fourier transformation and Fourier series?

Is there any connection between Fourier transformation of a function and its Fourier series of the function? I only know the formula to find Fourier transformation and to find Fourier coefficients to ...
3
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0answers
92 views

Reference request for Fourier analysis on local fields

I am studing Class field theory. I need a good reference books, notes etc. which explains the following topics : Ideles and ideals, haar volume measure and integration on local fields, Fourier ...
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0answers
59 views

Fourier transform of a Gaussian

I am trying to solve the following exercize: Show that Fourier transform of a Gaussian (a function of the form $Ae^{-\frac{x^2}{\sigma^2}}$) is also a Gaussian. So I did the required calculation (I ...
2
votes
4answers
367 views

Dirac Delta function inverse Fourier transform

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-i\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
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2answers
389 views

Fourier transform of $e^{-|t|}\sin(t)$

How can i calculate the Fourier transform of $e^{-|t|}\sin(t)$. I guess I need to do something with convolution, but I am not sure. Can somebody show me the way?
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17 views

Why don't we use unit impulse to find the fourier transform of unit step signal?

I have read that we can't find the fourier transform of unit step as it is not absolutely integrable. So we use signum function to find its transform .But why don't we use unit impulse function to ...
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1answer
696 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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1answer
42 views

Derive the Fourier Transform

I have been asked to derive the Fourier Transform for $$f(x)=\frac{1}{x^2+a^2}$$ where $a>0$. I know the Fourier Transform is equal to ...
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0answers
22 views

Cross-correlation, Fourier transform and Laplace transform: measure of how much signal are alike?

I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still ...
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2answers
35 views

Convergent Fourier series of continuous function

Let $f$ be a continuous function. It is known that its Fourier series is convergent almost everywhere to $f$ and it may fail to converge on some measure zero set. However I would like to know whether ...
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0answers
6 views

How can I obtain the inverse transform?

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to ...
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0answers
35 views

Bit operations to count longest string of 1s in a binary number - connections to FFT?

I found this rather applied question on another forum. How to calculate size of largest string of consecutive 1s in a binary number. However the other forum had neither much of a focus on applied ...
1
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1answer
37 views

trigonometric series

It is known that the eigenvalues of Sturm liouville problem: $$ u''(x)+\lambda u(x)=0 \\ u(0)=u'(\pi)=0 $$ are $\sin\left(\left(\frac{1}{2}+n\right)x\right)$ for $n=0,1...$ If for example we expand ...
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125 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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12 views

Can the DTFT be Pi-periodic?

Given is the following property of the DTFT of a time-discrete signal x[n]: $X(e^{j*\theta}) = X(e^{j*(\theta-\pi)})$ In my opinion this DTFT has a period of $\pi$ and not $2\pi$, as the definition ...
0
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1answer
22 views

about the property of Fourier transform??

It is said that: $$F[\frac{df(x)}{dx}] = i\omega F(\omega)$$. This expression depends on the initial definition of Fourier transform, yes? if I define Fourier transform as: ...