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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Integral of $\sin (e^{x^2})e^{-x^2+ix\lambda}$

Trying to solve this problem : Is $T$ invariant under Fourier transform ? Where : $T= \{f\in \mathcal{C}^{\infty} (\mathbb{R}), \forall n \in \mathbb{N}, |x|^nf(x) \to 0 \; \text{when}\; |x| \to ...
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1answer
37 views

Calculate $\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}$ using Fourier transformations

Calculate $\left(\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}\right)(y)$ using Fourier transformations. I have found a solution, but my method was very long. How could I shorten the solution? ...
2
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2answers
34 views

Calculating the convolution of a piecewise function

Let $$f(x) = \begin{cases} \frac{1}{2}, & \text{if $\rvert x\lvert \le 1$ } \\ 0, & \text{otherwise} \end{cases}$$ I want to calculate the convolution of $f$ with itself. I am ...
3
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1answer
245 views

The discrete Fourier transform of a Dirichlet charachter

I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language. I am trying to find the ...
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2answers
46 views

Calculating the Inverse Fourier Transform of $\frac{1}{\sqrt{2\pi}k}\sin k$

This used to be part of a longer question that I posted earlier but since that question seemed to long I decided to split it up. Given the function $$f(x) = \begin{cases} \frac{1}{2}, ...
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3answers
58 views

Show that the sum equals $0$ according to Cesàro

I'm stuck at some problems in my Fourier Analys course, maybe you got a clue. If $x \neq n \cdot 2 \pi$, then $$ \frac{1}{2}+ \sum_{k=1}^{\infty}\cos(kx) = 0 \tag{C,1} $$ Solution: So I know where ...
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13 views

Quesion about Parsevals formula for Fourier-Legendre Series

Question: A function $f(x)$ defined on $(-1,1)$ can be expanded as $$ f(x) \backsim \sum_{n=0}^{\infty} c_nP_n(x) $$ What do Parsevals formula look like for this expansion? My solution: Ok so I ...
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1answer
29 views

Deriving the coefficients during fourier analysis

I'm self-studying Fourier transforms, but I'm stuck on a basic point about integration during the derivation of an expression for the coefficients of the Fourier transform. For a function of period ...
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24 views

Complex coefficient in Fourier series

Why can $$\sum_{n=1}^N a_n \frac{e^{2 \pi i n t} + e^{-2 \pi i n t}}{2} + b_n \frac{e^{2 \pi i n t} - e^{-2 \pi i n t}}{2i} $$ be written as $$ \sum_{n=-N}^N c_n e^{2 \pi i n t}$$ for some setting of ...
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0answers
15 views

Why is the square of an image not equvalent to taking the autoconvolution of an image in fourier space?

Al right I know that in order to multiply in the normal domain I have to take the convolution in Fourier domain but when I do so in matlab and invert the result then I come up with nothing but a lot ...
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1answer
24 views

Ill-posed integral equation problem using Fourier Transfom

By using the Fourier Transform, show that the following equation $$\int_{-\infty}^{+\infty} K(x-y) g(y) dy = f(x), \qquad -\infty < x < \infty$$ is ill-posed. For overcoming ill-posedness of ...
3
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1answer
26 views

How to find the Fourier's coefficient $a_n$ of the Fourier's series of $\sin(x)$ on $(0,\pi]$, $0$on $(-\pi,0]$

Considering $g(x)$, periodical with a period of $2\pi$ defined by \begin{equation*} g(x)= \begin{cases} 0 & \text{for $x \in (-\pi;0]$} \\ \sin(x) & \text{for $x \in ...
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1answer
39 views

Strange equation after derivative

Assume Fourier integral of this function $ f(x)= \left\{\begin{matrix} 1 , \left | x \right |< 1 & \\ 0 , \left | x \right |> 1 & \end{matrix}\right. $ is this: $$ f(x)= \frac{2}{\pi ...
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2answers
95 views

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform?

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform? I had an idea in my mind. To use the $\text{sinc}$ function and take its inverse Fourier Transform or something like that. ...
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1answer
27 views

How to show that the Fourier's series of $f(x)=x$ uniformly converges?

How to show that the Fourier's series of $f(x)=x$ uniformly converges? After finding its coefficient, I got: $$\sum\limits_{n=1}^{+\infty}\frac{2(-1)^{n+1}}{n}\sin(nx)$$ I showed the pointwise ...
2
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32 views

Pointwise version of Fejer's theorem (convergence of Cesaro means)

Prove a pointwise version of Fejer's theorem: If $f\in \mathscr{R}$ and $f(x+),f(x-)$ exist for some $x$, then $$\lim \limits_{N\to \infty}\sigma_N(f;x)=\frac{f(x+)+f(x-)}{2},$$ where ...
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1answer
22 views

Why are the Fourier's coefficient on $0,2\pi$ and $-\pi,\pi$ the same?

I was given the Following Fourier's coefficient and I was happy with it: $$\left\{ \begin{array}{ll} a_n(f)=\frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos\left(nx\right)\,\mathrm{d}x\\ b_n(f) = ...
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1answer
31 views

Inverse Fourier-Stieltjes transform of $1$

Let $S(x) = \text{sgn}(x)/2$ for $x \ne 0$ and $S(x) = 0$ for $x = 0$. Then its Fourier-Stieltjes transform is $\hat{S}(k) = \int_{-\infty}^\infty e^{i k x} dS(x) = 1$. I tried to evaluate the ...
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1answer
29 views

Dirichlet theorem and expansion of fourier series

Dirichlet's theorem says that any function $f(x)$ on the interval $[-a,+a]$ can be expanded as a Fourier series: $$f\left ( x \right )=\sum_{n=0}^{\infty}\left [ a_{n} \sin \left ( \frac{n\pi ...
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3answers
3k views

Why fourier transformation use complex number?

I know that the Fourier transform is as follows:$$\hat{f}(\xi)= \int_{-\infty}^{\infty}\exp(-\mathrm ix\xi)f(x)\mathrm{d}x$$ but I couldent understand why should use complex number $i$ in the ...
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1answer
25 views

Existence of $L^{1}(\mathbb{R}^{n})$ Function Defined via Functional Equation

Perhaps, I'm reading the problem statement wrong, and it's not asking for existence, only uniqueness; but in any case... Problem. Let $g\in L^{1}(\mathbb{R}^{n})$, $\|g\|_{L^{1}}<1$. Prove that ...
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1answer
16 views

Fourier transform taylor series expansion

Given that $$\varphi_X(\xi)=\hat{f}_X(\xi) = \int_{-\infty}^{\infty}dx f_X(x)e^{i\xi x}$$ where $\xi = 2\pi v$, I perform TSE on $e^{i\xi x}$ (around $0$) obtaining that $$e^{i\xi x} = ...
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1answer
30 views

Magnitude of a complex exponential

I have this DTFT of an impulse response in a DSP course $H(\omega) ~=~ \sum_{n=-\infty}^{\infty} h[n]\, e^{-j \omega n} ~=~ \sum_{n=0}^\infty \alpha^n\, e^{-j \omega n} ~=~ \sum_{n=0}^\infty ( ...
3
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2answers
50 views

Fourier transform of signal $t \sin^2(t)/(\pi t)^2$

Does someone know how to do the Fourier Transform of the signal $$x(t) = t \cdot \frac{\sin^2(t)}{(\pi t)^2}$$ My first thought was: $$x(t)= \frac{t}{\pi^2} \cdot \frac{\sin^2(t)}{t^2} = ...
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1answer
37 views

Construct a nonnegative nonzero Schwartz function whose Fourier transform is nonnegative and compactly supported.

I tried the exercise with the hint that $\phi(x)=|\varphi\star\hat{\varphi}|^2$ could be the solution with $\varphi$ compactly supported and odd. Thus, \begin{align*} ...
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22 views

Fourier Transform of triangle function

I have a question regarding the FT of the triangular function: How does $e^{-j\omega t}$ becomes the cosine function in the first line? What happened to the sine when you go from $e^{j \omega t}$ ...
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2answers
40 views

Inequalities in proof of Bernstein-type lemmas

I'm working through the proof of Lemma 2.1 in Bahouri-Chemin-Danchin's Fourier Analysis and Nonlinear PDE. The place where I'm stuck boils down to (I think) proving the following inequality: ...
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1answer
38 views

Evaluating $S_n= \int_{\mathbb{R}} \left(\frac{\sin x}{x}\right )^n dx$ for $n=3$ using $n=2,4$ and the Fourier transform

Given that $S_n= \int_{\mathbb{R}} \left(\frac{\sin x}{x}\right )^n dx$ one can deduce the value for $n=2$ by considering the function $f=1_{[-1,1]}$ and applying Plancherel's theorem. Similarly for ...
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423 views

Reference Request: Texts on Fourier Analysis with emphasis on Number Theory

This may not exist but I would just like to ask in case.. I think Rudin is probably the best book but it doesn't have any (?) number theory in it. What is a good textbook (or comprehensive lecture ...
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1answer
33 views

Relationship between decay of Fourier transform and smoothness in $L^2$

This is Ex $2.3.6$ in Dym and Mckean's Fourier Series and Integrals. One can define the operator $\hat{f}(\gamma)$ for $f \in L^2(\mathbb{R})$ as $\lim_{b \rightarrow \infty, \, a \rightarrow ...
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16k views

Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas ...
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66 views

Fourier transform

I got confused with the following question when studying Fourier analysis: If there's a function $f\in L^2(\mathbb{R})$, with the Riemann integral $\int f(t) e^{-2 \pi it \gamma} dt \in ...
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1answer
106 views

Prove that $f(x) = 0$ on the interval $[0,1]$ [duplicate]

The function $f(x)$ is continuous on the interval $[0,1]$ and we have that $$ \int_0^1f(x)x^ndx = 0, \quad\, n=0,1,2,\dots $$ Prove that $f(x)= 0$ on the interval $[0,1]$. So I am thinking ...
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2answers
30 views

Use Fourier transform to solve the integral

Let $f(t)=1-t^2$ , for $|t|<1$ and $0$ elsewhere. Compute the Fourier transform of $f(t)$ and use the result to find the value of the integral $$ \int_{-\infty}^{\infty}\frac{\sin t-t \cos ...
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1answer
88 views

A Fourier multiplier mapping $L^{\infty}(\mathbb{T})$ into $C(\mathbb{T})$ corresponds to a function from $L^1(\mathbb{T})$

How can I prove that a Fourier multiplier sequence $\lbrace{m_n\rbrace}_{n=-\infty}^{\infty}$ mapping $L^{\infty}(\mathbb{T})$ into $C(\mathbb{T})$ corresponds to a function from $L^1(\mathbb{T})$? ...
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1answer
51 views

How can I find the Fourier transform of constant value like $1$.

The textbook told me that $\mathbb F[1] = \delta(f)$ and $\mathbb F[\delta(t)]=1$. It is easy to prove that $\mathbb F[\delta(t)] = 1$. $$ \mathbb F[\delta(t)] = \int_{-\infty}^\infty ...
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1answer
15 views

Short-Time Fourier Transform - why does the index range from negative to positive infinity?

I'm new to Fourier Transform. Could anyone explain to me in the Short-time Fourier Transform Equation (wikipedia): $$STFT\{x[n]\}(m,\omega) \equiv X(m,\omega) = \sum_{n = -\infty}^\infty ...
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15 views

Show that it is a positive summation kernel

Let $$ K(x) = \frac{3}{4}(1-x^2) $$ for $|x|<1$ and $0$ elsewhere Show that $$ K_n(x):=nK(nx) $$ is a positive summation kernel. Solution: So I want to show that $$ K_n(x)≥0 \\ ...
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1answer
31 views

What is the Fourier series of $f(x)$

What is Fourier series of $$f(x) = \sum_{n=1}^\infty \frac{\cos nx}{2^n}$$ Now, it was claimed that since $f(x)$ converge uniformly and: $$f(x) = \sum_{n=1}^\infty \frac{e^{inx} + ...
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0answers
36 views

Why does inputting complex exponentials into a system give its frequency response?

Let's say I have an FIR filter with the equation: $$ y[n] = \sum_{i=0}^{N-1} h[i] x[n-i] $$ I know that to find the frequency response of this filter, I need to input a complex exponential in place ...
4
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51 views

Hilbert space in Papa Rudin

In Rudin's Real and Complex Analysis, there is a problem in Chapter 4 on a Hilbert space $X = \text{span} \{e^{ist} \, \mid \, s \in \mathbb{R}\}$ with the inner product $$(f,g) = \lim_{T \to \infty} ...
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1answer
21 views

Fourier Transform of Pulse Train

I want to derive the Fourier transform of the impulse train. So far I have gotten up to this point. $$p(t) = \sum_{n = -\infty}^{\infty}\delta(t - nT_s)$$ $$P(\omega) = ...
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31 views

Can downsampling create energy at the Nyquist frequency?

I am a bit surprised by the following and would like to share it with you. I expect I am mistaken somewhere and will be happy to be corrected. I have searched StackExchange not only in Mathematics ...
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20 views

Sawtooth Waves that grow in magnitude

Are there functions that generate sawtooth waves that grow in magnitude?
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57 views

Solving Fourier Integration of $ f(x)=\begin{cases} \sin(x)&0\leq x\leq\pi\\ 0 & \text{remaining} \end{cases}$

I tried to compute Fourier integration of below function : $$ f(x)=\begin{cases} \sin(x)&0\leq x\leq\pi\\ 0 & \text{remaining} \end{cases} $$ my solution ended up as follows but I need a ...
3
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0answers
31 views

Lebesgue measurable with two periods

I am trying to prove that a Lebesgue Measurable function with two periods $a $ and $b$ such that $b/a $ is irrational must ne constant almost everywhere.... I really dont know what to do, it says that ...
3
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1answer
39 views

Fourier transform of simple function

I'm stuck on this problem Calculate the Fourier transform of $$ f(t) = \sin t \, , \, |t|<\pi \\ f(t) = 0 \, , \, |t|≥ \pi $$ So the start is really simple $$ \hat{f}(\omega) = ...
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1answer
54 views

Solving Fourier series of $ f(x)=\begin{cases} x+1 ;-1<x<0\\ 1-x;0<x<1 \end{cases} $

Please take a look at below Fourier series : $ f(x)=\begin{cases} x+1 &-1<x<0\\ 1-x & 0<x<1 \end{cases} $ I tried to solve it as follows : $ a_n=\displaystyle ...
0
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1answer
29 views

Parition of unity argument in a Fourier analysis paper

I am currently reading a paper "The proof of the $\ell^2$ decoupling conjecture" by Bourgain and Demeter. I was wondering if someone could explain me one of the arguments in their paper. First I will ...
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14 views

How does the weighted superposition of irreps make the Fourier transform of a finite group unitary?

This is supplementary to this question. In the lecture note of Andrew Childs on Nonabelian Fourier analysis, it is said that the Fourier transform of a finite group is the weighted superposition of ...