Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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23 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 $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
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1answer
38 views

Coefficient calculation on Fourier series !?

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}+ ...
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0answers
16 views

Proving and Deducing a Fourier Series [on hold]

Prove that in and deduce that I tried solving this using the Parseval's Theorem, but it couldn't be proved. I am also attaching the ways in which I tried to solve this..
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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 ...
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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 ...
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0answers
25 views

Consider fourier transformations of $|p(\mathbf{r})|^2$

If we have $\mathbf{k}=(k_x,k_y,k_z)=\frac{2\pi}{L}(j,s,l)$ with $j,s,l \in \mathbb{Z}$ and we have $$p(\mathbf{r})=\sum_{\mathbf{k}}\tilde{p}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}} \implies ...
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2answers
28 views

Complex Fourier coefficients for $e^{|x|}$

I'm new to Fourier expansions and transforms, and I'm not sure how to proceed with this question. I know a function f(x) can be expressed as an infinite sum of $c_ne^{in \pi x/L}$, and that $c_n = ...
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0answers
10 views

Trying to find the Fourier series of $f(x)$, where $f(x)$ is a piecewise function that includes $E\;sin(\omega\;t)$.

Here's the full function I'm trying to find the Fourier series to: $$f(x) = \left\{ \begin{array}{lr} 0 & : -\frac{\pi}{\omega}\leq t\lt 0 \\ E\;sin(\omega t) & : 0\leq ...
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0answers
14 views

Show that the convolution of the two time domain functions satisfy the relationship Y(q) = H(q) * U(q).

The convolution of two time domain functions h(t) and u(t) is given by $$ y(t) = \int_{-\infty}^{\infty} h(t- \tau)u(\tau)d\tau $$ Show that the Fourier Transforms Y(q), H(q) and U(q) satisfy the ...
4
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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}< ...
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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 ...
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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 ...
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0answers
22 views

Fourier transform turns product into convolution [on hold]

Suppose $f,g \in L^2(T)$, show that the fourier coefficients of $ fg \in L^1(T)$ are given by the formula $\widehat{fg}(\xi)=\sum \hat{f}(k)\hat{g}(\xi-k)$ Thank you for your help!
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1answer
31 views

Hard Integral [Heat Equation + Fourier Sine Series]

I encountered this integral while doing a heat equation problem in Advanced Calculus. How does the person evaluate the integral involving $$\int_0^\pi \sin x \cos (nx) \: dx $$ Can someone ...
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1answer
22 views

Fourier transform of a scaled variable [duplicate]

If $f\hat(k)$ is the fourier transform of $f(x)$, what is the fourier transform of $f(x/c)$ where $c$ is a real number greater than $0$?
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35 views

Fourier Series of $f(x) = x + |x|$ [on hold]

Compute the Fourier series of $f(x)=x+|x|,-\pi\le x\le \pi$ Please help on solving the equation.
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0answers
9 views

Discrete Fourier Series Transformations

Let the DFT of f[n] be given by F[k]. Find the DFT G[k] of time series g[n] = f[n] * (-1)^n in terms of F[k]. I know that G[k] is related to F[k] by a shift in the frequency domain, but I'm not ...
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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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29 views

asymptotics of the Fourier transform of Daubechies wavelet

I want to evaluate the series \begin{equation} S(\alpha,\omega)=\sum_{k=-\infty}^{\infty}\frac{|\Psi(2k\pi-\omega)|^2}{|2k\pi-\omega|^\alpha} \end{equation} where $0\le\omega<2\pi$, ...
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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 ...
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56 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
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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 ...
2
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0answers
37 views

Fourier Series of the batman equation

I want to represent the batman equation as a Fourier Series. (I got the equation here : Is this Batman equation for real?) But a part of it is an ellipse and when I tried to calculate an the integral ...
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1answer
18 views

Complex Fourier Series of $t^3$

I am trying to find compute the complex Fourier series of the following function: $$f(t) = t^3$$ $$-\frac32 \le t \le \frac32$$ $$f(t) = f(t+3)$$ I am using the generic function for the complex ...
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1answer
35 views

The Fourier transform of exp(-x)*heaviside(x)

I'm trying to understand the Fourier transform of Y=exp^-x. Since the term tends to -infinity I have to multiply Y by the heaviside function to set everything below 0 to 0 so I can successfully ...
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0answers
26 views

find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
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1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
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0answers
21 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
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1answer
15 views

Find the Fourier Transform of piecewise finction

$$f(x) = \begin{cases} 0 & |x|> a \\ 1 & |x|< a \end{cases}$$ I have most of the solution, I'm just faltering on obtaining the sin(ax) part of the solution, I'm missing an exponential ...
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3answers
26 views

How do I find the solution to this summation after computing the following power series?

I have found that the Fourier cosine series from $({-\pi},{\pi})$ of the function $f(x)=\cosh(x)$ is $$ \frac{2\sinh({\pi})}{\pi}\left[\frac{1}{2}+ \sum_{n\: =\: 1}^{\infty}\:\ ...
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2answers
49 views

Integral using Parseval's Theorem

How would I integrate $$\int_{-\infty}^{+\infty} \frac{\sin^{2}(x)}{x^{2}}\,dx$$ using Fourier Transform methods, i.e. using Parseval's Theorem ? How would I then use that to calculate: ...
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1answer
37 views

Solve differential equation using fourier series

I am trying to solve this problem in my analysis book in a chapter on Fourier series: Solve the differential equation $$(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) u(x,y) = ...
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0answers
12 views

Chladni patterns

So I was watching this video on Chladni figures (https://www.youtube.com/watch?v=wvJAgrUBF4w) and thought that it would be nice to replicate a few of these, especially the more complicated, high ...
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0answers
23 views

Table of Fourier series

I found that there are very good references on Fourier integral transform but none on Fourier series. Do you happen to know one?
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1answer
14 views

ODE with finite Fourier expansion periodic coefficients

Regard the ordinary differential equation $$ \dot a(t) = z(t) a(t) $$ where $a(t)$ and $z(t)$ are matrix valued such that $z$ is periodic ($z(t+2\pi)=z(t)$). Then it is well-known (Floquet theory), ...
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0answers
11 views

Compute the Fourier series of a piecewise function.

Consider the function: $ f(\theta) = \begin{cases} 0 & \text{if } |\theta| >\delta \\ 1-|\theta|/\delta & \text{if } |\theta| \leq \delta \end{cases} $ I need to show ...
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0answers
34 views

Show that the Fourier series is $\frac{8}{\pi} \sum_{k \;odd \ge 1} \frac{sin(k \theta)}{k^3} $

Consider the odd function $f(\theta)=\theta (\pi - \theta)$, then I need to show that: $f(\theta)=\frac{8}{\pi} \sum_{k \;odd \ge 1} \frac{sin(k \theta)}{k^3}$ then I computed the Fourier ...
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2answers
37 views

Show that $\widehat{f}(n)$ is zero for odd $n$

The following problem is from Stein´s Introduction to Fourier analysis: Suppose that $f(\theta + \pi)=f(\theta)$ for all $\theta \in \mathbb{R}$ Show that $\widehat{f}(n)$ is zero for odd $n$. My ...
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1answer
26 views

Writing a Fourier series of a $2\pi$-periodic function.

This problem was taken from Stein's Introduction to Fourier analysis, and it goes like this: Let $f$ be a $2\pi$-periodic Riemman integrable function defined on $\mathbb{R}$. Show that the Fourier ...
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13 views

Fourier Series from a simple sinosoidal function

I have a function of: Vd = (Rd/(33e3+Rd))*(Vrf*sin(Wrf*t)+Vlo*sin(Wlo*t)); y_t = 11*Vd; func_ = y_t; where: ...
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2answers
50 views

A difficult trigonometric integral involving absolute value

$$ \int_{0}^{2\pi}\lvert\sin(x)\rvert\cos(nx)\,dx= -\frac{4\cos^2\bigl(\frac{\pi n}{2}\bigr)\cos(\pi n)}{n^2-1} $$ I'm not actually trying to solve this myself. The answer appears in my lecture notes ...
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1answer
66 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
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1answer
24 views

Do I have to transform the solution into $u(x, y)$?

Find the solution of the problem $$u_{xx}(x,y)+u_{yy}(x,y)=0, x^2+y^2>1 \\u=1+3\sin^3 \theta , 0 \leq \theta <2\pi$$ $u$ is bounded. I have done the following: $$u(x,y)=v(\rho, \theta) \\ ...
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17 views

Can a modular form every have a polynomial expansion in q?

In other words, if q = e^(2*piiz), where z is in the upper half plane, can the fourier expansion of a non-constant modular form ever have a finite number of terms?
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1answer
30 views

Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
3
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1answer
38 views

$\|fg\|_{A (\mathbb T)} \leq C \|f\|_{L^{2}} \|g\|_{A (\mathbb T)}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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1answer
122 views

Is there a closed-form of $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^4}$

Is there a closed-form summation result for Fourier series: $$\sum_{n=1}^{\infty}\frac{\sin(n)}{n^4}?\tag{1}$$ I tried using available result of the following (odd) function : ...
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1answer
30 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
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0answers
20 views

Properties of Fourier coefficients of real valued functions

Let $\hat{f}(n)$ be the Fourier coefficients of $f:[0,2\pi]\to \mathbb{C}$ defined as $$\hat{f}(n)=\int_{0}^{2\pi}f(x)e^{-{\rm{i}}nx}\,\mathrm{d}x$$ Note $f$ is Riemann-integrable on $[0,2\pi]$. We ...
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0answers
9 views

Finding Fourier coefficients of (discrete ) $cos(\frac{6*n*\pi}{N})$

What is the Fourier coefficients of (discrete ) $cos(6*n*pi/N)$? The answer says $0.5[delta(k-3)+delta(k+3)]$ (delta is Dirac delta function)...my attempt was to use a formula $1/N(sum from 0 to ...