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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Fourier Series of Complex Valued Functions

Write the Fourier series of functions in the space of complex valued functions L2([0; 1]), which we view as periodic functions on R. Specify the coefficients of the expansion and also express them as ...
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1answer
21 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
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12 views

Numeric Evaluation of Double Surface Integral over Greens Function with Singular Points

I'm currently using python to numerically evaluate the follow expression ...
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1answer
11 views

Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
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1answer
26 views

Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
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21 views

Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
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2answers
27 views

Getting fourier coefficient by integrating over half the period?

In the book Schaum's Outlines of Analog and Digital Communications solved problem 1.2, the author calculates the fourier coeffecient $C_0$ for the rectangular pulse train: where $a$ is assumed to be ...
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1answer
15 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
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1answer
35 views

Difficult integration

In my notes the lecturer takes the Fourier transform in $x, y$ and $t$ of $\phi(x,y,z,t)$ as: $$ \int_{-\infty}^{\infty}dt\, e^{i\omega ...
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11 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
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2answers
78 views

Is Fourier transform still writing a function as a series of sines and cosines?

In the Fourier series we write a function as a series of sines and cosines. Fourier transform seems to me to be totally different, we are not finding a series but rather a function $\hat f(w)$. So ...
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1answer
23 views

Fourier series and evaluation of another series

I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula: ...
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1answer
18 views

What kind of information is available in a Fourier series expansion of an analytic function that is not (readily) available in a Taylor series?

What kind of information is available in a Fourier series expansion of a real analytic function that is not (readily) available in a power series? When would one know to work with one over the other?
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32 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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2answers
99 views

Coefficient calculation on Fourier series !? [on hold]

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
20 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
15 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
17 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
26 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
30 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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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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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 ...
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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 [closed]

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

Compute the Fourier series of $f(x)=x+|x|,-\pi\le x\le \pi$ Please help on solving the equation.
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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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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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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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23 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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58 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
44 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 ...
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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
36 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
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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
39 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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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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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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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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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 ...