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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Relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring for dummies?

As the question title suggests, what is an explanation for dummies of the relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring?
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132 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall x \...
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1answer
40 views

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$?

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$ I have a proof for the first case, under the assumption that $f$ is $C^1$ and real valued (also $1$ periodic) $\lVert ...
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1answer
36 views

The Fourier series of a continuously differentiable function converges to it pointwise

$S_N(f):=\sum\limits_{|n|\le N}\hat f_n\cdot e^{i2\pi nx}$ where $\hat f_n=\int_0^1f(x)e^{-i2\pi nx} dx$ and $f$ a $1$-periodic $C^1(\mathbb R,\mathbb C)$ function, then $S_N(f)$ converges pointwise ...
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2answers
37 views

If $A(x)$ is 1-periodic and $\frac{A''(x)}{A(x)} = C$, then $C=-4\pi^2 n^2$?

This might be a trivial question but I forgot my differential equation. Anyway, I am trying to solve the heat equation on circle. Given that $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\...
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1answer
31 views

Can Mathematica/WolframAlpha do a Fourier transform for f instead of ω?

When Mathematica/WolframAlpha calculates the Fourier Transform, it calculates it using the angular frequency. How do I make the Fourier transforms Mathematica/WolframAlpha to match the following table?...
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1answer
58 views

If a fourier series converges to an elementary function, can I then find the closed form of this function?

Suppose that I am told that f(x) is some elementary function and that f(x) has the fourier series $\Sigma_{k=-\infty}^{\infty}c_ke^{ikx}$. By "elementary function" I mean: https://en.wikipedia.org/...
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1answer
1k views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, k\...
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1answer
39 views

Fourier Analysis / Real Analysis

I have run into the following exercice from Grafakos' Classical and Modern Fourier Analysis: if $f$ is a function in $L_{1}(R)$, then one has to prove that $\int_{-\infty}^{\infty} f(x)dx$ = $\int_{-\...
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35 views

Derivative of Fourier transform with respect to intermediate variable

I am studying a system with a characteristic, say $\zeta$, that varies in 3D real space. I can use this characteristic to calculate the value of a second characteristic $\beta$. In other words, I have ...
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0answers
21 views

Fourier Convolution Inversion

Consider a Fourier convolution $f(x) = (g * h)(x)$, where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ...
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2answers
855 views

Fourier series is to Fourier transform what Laurent series is to …?

Since the coefficients $$a_k = \frac1{2\pi i}\oint_C\frac{f(z)}{(z-c)^{k+1}}\,dz$$ for the Laurent series $$f(z)\Big|_{r\le|z|\le R} = \sum_{k=-\infty}^{\infty}a_k\cdot(z-c)^k $$ of a function $f\...
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0answers
31 views

Suppose $\sum |A_n|^2 <\infty$, then $\sum r^{|n|} A_n e^{inx}$ converges uniformly?

Suppose $\sum_{-\infty}^\infty |A_n|^2$ converges. Show that for each $r\in (0,1)$, the series $\sum_{-\infty}^\infty r^{|n|} A_n e^{inx}$ converges uniformly in $x$. I know that the series $\sum_{-\...
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1answer
98 views

Parseval's identity holds

Theorem: If $u \in L^2(\mathbb{R}^n)$ then the Fourier transform $\widehat{u} \in S'(\mathbb{R}^n)$ is a $L^2(\mathbb{R}^n)$ function and the Parseval's identity holds: $||\widehat{u}||_{L^2(\mathbb{R}...
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0answers
34 views

Result of a decay condition

Assuming that a function g is such that $ g(x) \leq C ( 1 + |x|)^{(-1 - \varepsilon)}$ for some $\varepsilon > 0$ , then how can we prove that $ \sum_{n = - \infty}^{n = + \infty} | g(x- k - \frac{...
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1answer
73 views

If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges

I'm learning about Fourier series, specifically Cesàro summable sequences and series, and need help with the following problem: Show that if the series $\sum_{k=1}^{\infty} a_k$ is Cesàro ...
2
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2answers
77 views

Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$

Show that $\int_{-\infty}^\infty f(w)e^{-\pi \delta w^2} dw \to \int_{-\infty}^\infty f(w) dw$ as $\delta \to 0$. $f(w)$ is a Schwartz function. This is a part of the proof of Fourier inversion ...
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0answers
19 views

Integration on $R^d$ about Changing Coordinate

I have the formula $$\int_{\mathbb R^d} F(x)dx=\int_{S^{d-1}}\int_0^\infty F(r\gamma)r^{d-1}drd\sigma(\gamma)$$ Problem: Apply this to $F(x)=g(r)f(\gamma)$, where $x=r\gamma$, to prove that for ...
2
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2answers
177 views

About the Fourier-Legendre series of $f(x)=e^{-x}$

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} \int_{-1}^{1}\...
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6answers
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Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
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1answer
50 views

Fourier series of piecewise-defined function and convergence

I'm learning about Fourier series and need help with the following problem: Consider the function $$g(x) = \begin{cases} x^{\frac{1}{3}}, & x \in [0, \frac{\pi}{2}] \\ (-x)^{\frac{1}{4}}...
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1answer
38 views

If $f: \mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic, then $f \in L^2[-\pi, \pi]$

I'm learning about Fourier series, specifically $L^2$ convergence, and need help with the following problem: Let $f: \mathbb R \to \mathbb R$ be continuous and $2\pi$-periodic. Show that $f \in L^...
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2answers
39 views

Proving, that $\text{Arg}(-i\sin(x))=\pi/2\text{sgn}(x)$ on $(-\pi,\pi)$

Alright. I thought, that $\text{Arg}(-i\sin(x))=3\pi/2$, however, the Wolfram Alpha tells a different story. I am sure that it must be kind of true, because $\text{Arg}(\sin(x))$ is the result of sum ...
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1answer
31 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
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0answers
21 views

Relation of rate of decay of a function with width of peaks of its Fourier transform

Consider a function $f(t)=\theta(t)e^{-\sigma_0 t}\sin(\omega_0 t)$, where $\theta(t)$ is $1$ for positive $t$ and $0$ for negative $t$. Its Fourier transform can be easily computed. It has the ...
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44 views

Absolute maximum

I´m trying to find the absolute maximum of $(2N-1)$ partial sum of the Fourier´s series of signum function on $[0,\pi]$, I have: $S_{2N-1}[f](x)=\frac{4}{\pi}\displaystyle\sum_{k=0}^{N-1}{\frac{sen((...
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0answers
68 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: $f(x)=\frac{1}{\pi}\...
3
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1answer
413 views

About the order of the $L^1$ norm of the Dirichlet kernel.

Reading this text from Wikipedia, I found the following statement about the Dirichlet kernel: $$\| D_n \|_{L^1} \approx \log n, $$ where $\approx$ denotes "is of the order". I think that this mean ...
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1answer
45 views

Can you recover a distribution from mollification?

Let $f\in \mathcal S'(\mathbb R)$ be a Schwartz distribution. Given $\rho \in C^\infty_c(\mathbb R)$ define the convolution as the function $$x\mapsto (f\ast\rho)(x):=\langle f, \rho (\cdot -x)\...
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1answer
53 views

Retrieving coefficients of polynomial from DFT.

I am trying to multiply two polynomials using DFT and I don't know how to get the last bit from the DFT of their multiplication. So there's $p(x) = x - 4$, DFT $-3$, $i-4$, $-5$, $-i-4$. And $q(x) = ...
5
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1answer
101 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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2answers
40 views

Finding the coefficients of a triangular wave.

I have the following equation that I want to solve $$a_k = \color{blue}{\frac{1}{T} \int_{0}^{T/2} 2 \frac{t}{T} e^{-i \frac{2\pi}{T}kt} dt} + \color{red}{\frac{1}{T} \int_{T/2}^{T} 2 \frac{T-t}{T} e^...
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2answers
996 views

Difference between Fourier integral and Fourier transform

What is the difference between Fourier integral and Fourier transform? I know that for Fourier integral the function must satisfy that : $\displaystyle \int_{-\infty}^\infty |f(t)| dt < \infty$, ...
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1answer
7k views

Fourier basis functions

What are Fourier basis functions? And how do I prove that Fourier basis functions are orthonormal?
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1answer
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Wiener filter: A good tutorial

I am interested in image analysis and am looking for an approachable tutorial to the Wiener filter. At some point I am interested in implementing such a filter but I would like to have a deeper ...
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1answer
38 views

calculate Fourier Transformate

i have the following exercice: Let for all $x \in \mathbb{R},$ $f(x)= \cos x$ and $g(x)= \sin x$. Calculate $T=f \delta' + g \delta''$ for this question, i find $T=3 \delta$. Calculate the Fourier ...
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2answers
54 views

Relation between Dirac function and inverse fourier transform of 1

According to my notes, it holds that $\delta=(2 \pi)^{-n} \widehat{1}$. How do we get the equality? We have that $\delta=\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \widehat{\delta}(\xi) e^{i x \xi} d{\...
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2answers
46 views

Is the Hilbert transform of a Schwarz function essentially bounded?

My ultimate goal is to show that the Hilbert transform of a Schwarz function is in $L^p(\mathbb{R})$, for every $p \in (1,\infty]$ (the definition I am using is $Hf(\xi) := \mathcal{F}^{-1}[(-i \ \...
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0answers
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Formula of phase correlation

If I have two 2D signals, and one is the shift of another. I can propose such schema for define offset via continious Fourier Transform: $$f_2(x,y)=f_1(x-x_0,y-y_0)$$ Then $$Ff_2(s_1,s_2)=e^{-2\pi j(...
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2answers
35 views

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
19 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 $1/k^n$, $n\geq0....
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0answers
33 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 \...
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1answer
29 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
26 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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1answer
46 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
18 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 $t=0$, $\mathcal{F}_R(t)=R$...
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0answers
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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26 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
36 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 $\...
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2answers
34 views

Help with an Inverse Fourier transform

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