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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Evaluate integral of $2\pi$ periodic function's multiplication

Let f and g be $2\pi$ periodic Riemann integrable funtions. I want to evaluate $\lim_{n \to \infty} \frac{1}{2\pi} \int_0^{2\pi}f(x)g(nx)dx$ I think that this integral express by using fourier ...
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27 views

Doesn't the recursive Fast Fourier Transform violate f(-x) =/= f(x) for odd functions?

When you recursively split into $Y_{even}$ and $Y_{odd}$, from the second recursion onwards don't these sets have their even-ness and odd-ness violated? I.e., assume you are running the FFT algorithm ...
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237 views

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=...
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24 views

Homogeneous solution, Fourier mode, wavelength

Consider a system $$ \partial_t u=N(\sigma)u,~~u=(x,t)~~~(1) $$ where $N$ is a non-linear operator depending on some control parameter. Suppose that the system (1) admits a homogeneous ...
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119 views

Averaging transformation of a closed plane curve

Let's suppose we have a closed plane curve of some shape whose points are described by the single parametric equation $P(x(t), y(t))$ where $t$ is some increasing parameter (example circle) or by ...
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66 views

Why the range of time period of exponential Fourier series is different from other two types of Fourier series?

Trigonometric Fourier series is given as $$x(t)=\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T) \quad(1)$$ Polar FS is given as $$x(t)=...
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Decay of Fourier Coefficients

Everybody knows that the Fourier coefficients of an $L^1$ function converge to zero. For an $L^2$ function we can say much more: $(\hat f(k))_{k\in\mathbb Z}\in\ell^2$. Therefore, it is reasonable to ...
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54 views

Bound related to Schwartz space

If $u \in \mathcal{S}'(\mathbb{R}^n)$, then is there an integer $m \ge 0$ and $C>0$ such that for all $\phi \in \mathcal{S}(\mathbb{R}^n)$,$$|u(\phi)| \le C\|\phi\|_m,$$where$$\|\phi\|_m = \sum_{|\...
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36 views

Equidistributed problem about polynomial with irrational coefficient

This problem is from Stein, Fourier Analysis,Chapter 4,problem 2(d). Problem:Suppose that $P(x)=c_n x^n+……+c_0$ is a polynomial with real coefficients, where at least one of $c_1,……,c_n$ is ...
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39 views

Calculating a uniform-in-wavelength Fourier transform

If $f$ is frequency in Hertz and we assume $c = 1$, then $\lambda = 1/f$ and we have the representation$$s(t) = \int_{-\infty}^\infty \hat{s}(1/\lambda)\exp(i2\pi t/\lambda)1/\lambda^2\,d\lambda.$$How ...
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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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37 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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29 views

Proof check: Isoperimetric inequality

My Proof: Let the arc-length parametrization of the curve be $\gamma(s) = \langle x(s),y(s)\rangle$. By Green's Theorem, the area $\mathcal{A}$ is $$ \mathcal{A} = \int_{0}^{2\pi} -y(s)x'(s) dx$$ ...
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32 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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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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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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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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31 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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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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19 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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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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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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75 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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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 ...
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72 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 ...
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44 views

If $f \in L^2(R)$, then $\hat{f}\in L^2(R)$?

The Fourier transform of $f$ is as follows: $$\hat{f}(\omega)=\frac{1}{‎‎‎\sqrt{2‎\pi‎}}\int_{-\infty}^{+\infty} e^{-i\omega t}f(t)dt$$ I need to know that if $f \in L^2(R)$, then can we conclude that ...
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55 views

If $f \in L^1(\mathbb{R})$, then $\hat{f}(\omega)‎\to0$ as $‎\omega‎ \to+‎\infty‎$?

The Fourier transform of $f$ is as follows: $$\hat{f}(\omega)=\frac{1}{‎‎‎\sqrt{2‎\pi‎}}\int_{-\infty}^{+\infty} e^{-i\omega t} f(t) \, dt.$$ I need to know that if $f \in L^1(R)$, then can we ...
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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^...
3
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1answer
48 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
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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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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20 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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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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1answer
95 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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1answer
42 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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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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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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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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45 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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43 views

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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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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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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34 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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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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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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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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25 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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2answers
33 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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28 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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1answer
33 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 $\...