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

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1answer
43 views

Where does the $2 \pi$ come from in the Fourier Transform Equation?

So I was working through the Fourier transform equations that arise. I was wondering where the radical outside the integral originated from? $\hat{f}(k) = \sqrt{\frac{|b|}{(2 \pi)^{1-a}}} ...
4
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2answers
58 views

Closed form of a series (dilogarithm)

We are all aware of the dilogarithm function (Spence's function): $$\sum_{n=1}^{\infty} \frac{x^n}{n^2}, \;\; x \in (-\infty, 1]$$ Also it is known that: $$\sum_{n=1}^{\infty} \frac{\cos n x}{n^2}= ...
2
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1answer
44 views

How to prove that this series converges uniformly?

I have a series $$ -\frac{\pi}{12} + \sum_{k=1}^\infty \frac{\left(3k\pi^2-16\right)\sin{\frac{k\pi}{2}} + 8\pi\cos{\frac{k\pi}{2}}}{\pi^2k^3}\cos{kt} $$ And I have to use Weierstass test to prove ...
2
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2answers
22 views

Calculating values of integrals using Fourier series and uniform convergence

I have a problem that I don't know how to begin solving. I have f(t) $$ f(t) = \sum_{k=1}^\infty\frac{1}{k^2+1}\sin{kt} $$ First I had to show that this series converges uniformly, I've done that ...
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0answers
20 views

Why is the period of the function you are generating a fourier series for important?

I try to visualize this in terms of something analogous to vector spaces in linear algebra because thats the only way I can understand the fourier series. You have a basis {$cos(0x), sin(0x), cos(x), ...
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0answers
23 views

Can you define a Fourier series for $f(x)$ for any interval you want, even for which $f(x)$ is not periodic on that interval?

Say you have a function $f(x)$, and you generate a Fourier series for it on the interval $-L \leq x \leq L$, and $f(x)$ is piecewise continuous on that interval. Say also that $f(x)$ is a periodic ...
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0answers
17 views

Finding $a_0$ for the function $s(t)=1-e^{-2t}$.

I am working on multiple Fourier series questions about the function $s(t)=1-e^{-2t}$. How do I find a naught as in $a_0=\dfrac 1T\displaystyle \int\limits_{t_0}^{t_0+T}s(t)\,dt$, when $T = 3$? ...
0
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1answer
38 views

How to derive the complex Fourier series of $s(t) = 1-e^{-2t}$? [on hold]

I have the periodic function $s(t)=1-e^{-2t}$. I am required to derive the complex Fourier series of $s(t)$. I have some knowledge of Fourier series but not enough to know if I am doing it correctly. ...
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0answers
32 views

Fourier series for a Sinusoid in a conventional way?

So my TA in class introduced this amazing way of finding fourier series coefficients for a sin wave, by writing $ sin( \omega t ) = (e^{i\omega t}-e^{-i\omega t}) / 2i $ ----(1) Hence getting the ...
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1answer
18 views

Can a fourier series generated for a function f(x) on some interval be used to calculate the value for f(x) for all x, even outside that interval?

If you have fourier series for a function f(x) on some interval a < x < b. Does it series still converge to the value of f(x) even for x that is not in that interval?
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1answer
22 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
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1answer
33 views

Calculate complex Fourier coefficients

I had this question in my end of year exam and am reviewing it before my supplemental paper. Calculate the complex Fourier coefficients $C_n$ for $$f(x)= |x|- \dfrac{\pi}{2} , |x|\leq \pi $$, and ...
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0answers
22 views

Fourier methods and a conductor bar

I was doing this question bellow: I tried: Could you help me in the 3 (second Picture) and how to solve the problem?
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2answers
78 views

Prove the Identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $

By considering the fact that $f(\pi/2)=1$, prove the identity $\pi=2- \sum_{1}^{∞} \frac{(-1)^m}{m^2-\frac{1}{4}} $ This question was is a subsection in a chapter on Fourier series, can I use my ...
1
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1answer
33 views

Methods for solving definite trig. integrals?

I am studying Fourier series and there is a lot of integration going on, specifically with trigonometric functions involved. When solving for the Fourier coefficients, often times, the definite ...
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2answers
31 views

How to do Fourier transform for these 2 questions?

I don't get certain of parts of these two questions 1) I'm trying to do the Fourier transform of: $$f(x) = \, xe^{-x^2} $$ In the problem it said to use: $$F \, (e^{-tx^2}) = ...
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0answers
28 views

Cosine Fourier series solution of semi-major axis nonlinear integral equaton

Consider an integral equation $$ \frac{1}{z(t)}=f(t)+\alpha\int_0 ^\infty \cos(ts)z(s)\,ds $$ I am required to solve for $z(t)$. I approached this problem by considering the integral on right hand ...
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0answers
32 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
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2answers
26 views

Finding whether a piecewise function is even or odd

A periodic function with period $2\pi$ is defined by $f(x)=1$ in the interval $ a\lt x \lt b$ and $f(x)=0$ elsewhere. Can the function be even or odd? If not why not and if so, for what values of ...
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0answers
54 views

Calculate the Fourier series in complex exponential form

I'm studying for a test next week and came across this question in the past exam papers I've looked back in my notes, but I haven't a notion how to even attempt it. All help is appreciated. Calculate ...
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0answers
18 views

Proof about fundamental frequency for periodic functions

I know that in the Fourier series expansion for $f$, we have $$f(t) = \sum_{n=-\infty}^{\infty} c_n \exp\left(\frac{2\pi int}{T}\right)$$ where the lowest frequency term (ignoring the constant) has ...
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2answers
25 views

Fourier series on general interval $[a,b]$

Currently I'm studying Fourier series and the first thing I've read is the definition of the series for a function $f : [-\pi,\pi]\to \mathbb{R}$. In that case the Fourier series is ...
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0answers
46 views

Approximation by sinc functions in L2

I wish to find the best approximation in $ L_{2} (\Re )$ of $f(x)=\frac{sin(ax)}{ax}$ for $0<a<\pi$ and for $a>\pi$ , Using the system of sinc functions: $$g_{n}(x)=sinc(\pi x-\pi n) = ...
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1answer
31 views

Fourier analysis expansion $f(x) = \sin(x)$

I am reading a book on Fourier analysis and I am having difficulty in understanding a step in the expansion of the function $f(x) = \sin(x), 0 \lt x \lt \pi$ as a Cosine Fourier series. I attached ...
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0answers
12 views

sound FFT data to PCM data

I have only a quite naive understanding of FFT. Is my naive interpretation about how to recalculate the time series data (PCM data) back from FFT data correct? If not, what's wrong with it? I will ...
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0answers
14 views

Fourier Series Representation for Solid Vibrations in a Ball

I am trying to solve the following PDE eigenvalue problem for solid vibrations in a ball. \begin{cases} u_{tt}=c^2\Delta u&\text{in }D\\ u=0 &\text{on }\partial D\\ u=\phi (r,\theta) \text{ ...
2
votes
1answer
39 views

Why this formula doesn't work for $n=1$?

I've been studying Fourier series and in trying to compute the Fourier series for the function $f: (-\pi,\pi)\to \mathbb{R}$ given by $f(x)=|\sin x|$ I've found something quite strange that I'm not ...
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1answer
39 views

how to disprove uniform convergence

I've been asked to check the uniform convergence of the following function sequence on the real line: $$ f_{N}(t)=\sum_{n=-N}^{n=N}\sin(n) \,\frac{\sin(\pi t-\pi n)}{\pi t-\pi n} $$ It is asked in a ...
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1answer
34 views

Fourier series… delta notation

I'm going through the answer to a Fourier series question and have come across some notation which I haven't seen before. The question is to represent the periodic function $$f(x) = ...
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0answers
17 views

Help completing fourier series question.

I am currently solving a fourier series question and I've done the first 2 parts of the question which was to sketch the graph of the given function and also find the fourier series of $f(t)$. ...
2
votes
1answer
35 views

Fourier Expansion

A periodic function f(x) is defined by: $ f(n) = \begin{cases} {-x^2} & \textrm{ for - π < x ≤ 0} \\ x^2 & \textrm{ for 0 ≤ x < π } \\ \end{cases} \space , ...
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1answer
26 views

fourier series expansion show that:

I have a problem that I've partially worked but don't understand the next part/have made a mistake? $f(x)=0$ for $-\pi< x<0$ and $f(x)=x$ for $0≤x≤\pi$ I have $a_0=\dfrac \pi 4$ and $a_n=0$ ...
2
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2answers
27 views

Different approaches for evaluating a series

I was playing around with the Fourier series of $\cos x , \; x \in (0, \frac{\pi}{2})$ and a series came up. First of all the fourier series of $\cos x$ at the given interval is: $$\cos x ...
3
votes
1answer
57 views

Find this Limit (Fourier series)

Find the following limit: $$\large\lim_{n\to\infty} \int_{-\pi}^\pi \biggl(x + \frac{\pi}{2}\biggr)^2 \frac{\sin\bigl(\bigl(n+\frac{1}{2}\bigr)x\bigr) + x \cos nx}{\sin \frac{x}{2}}\,dx.$$ We tried ...
0
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1answer
46 views

Fourier transform of $f(x)=x$ if $0<x\leq 1$ and $f(x)=0$ otherwise

What is the Fourier transform of the function defined by $f(x)=x$ on $[0,1]$ and $f(x)=0$ otherwise, i.e., $\hat f(\xi) = \int_\mathbb{R} { e^{-iu\xi} f(u) du }$? Is there a closed-form? Else, how ...
2
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2answers
24 views

A proof of a known identity using Fourier series

The exercise Let $f$ be a $2\pi$ periodical function defined as $f(x)=\cos ax, \; |x|\leq \pi, \; a \notin \mathbb{Z}$. Expand $f$ in a Fourier series and prove that: $$\pi \cot \pi a = ...
4
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1answer
38 views

Fourier series expansion of $x(t) = \sum\nolimits_{z \in \mathbb{Z}} (-1)^z \delta(t - 2z)$

Find the Fourier series expansion of $x(t) = \sum\nolimits_{z \in \mathbb{Z}} (-1)^z \delta(t - 2z)$, where $\delta(\cdot)$ denotes the Dirac delta function (unit impulse). I can infer that the ...
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0answers
35 views

What functions stem from Fourier Series with rational-only coefficients?

Given the Fourier series $$f(z) = \sum_{k=-\infty}^\infty c_k e^{ikz}$$ but with $c_k\in(\mathbb Q+ i\mathbb Q)$ instead of $\mathbb C$ (or even purely real), are the functions obtained this way in ...
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0answers
90 views

Is there a closed-form approximation to a band-limited sawtooth?

A partial Fourier Series with no coefficients is equal to the closed form expression: $${A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - ...
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1answer
27 views

Strange inequality in the proof of differentiability of Fourier series

I am looking at a proof and I found a strange inequality. Let $n\in \mathbb{Z}^d$ then it is stated that $\sum_{j=1}^d{(2\pi)^{2k}n_j^{2k}}>>\parallel n\parallel_2^{2k}$ due to the inequality ...
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0answers
63 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting:\ $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
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vote
1answer
35 views

Find the Fourier series

Could somebody please tell me if I've gotten this question correct? I'm unsure about my answer. Consider the periodic function: $$f(x)= \begin{cases} 0,\ -\pi \lt x \le 0\\ 1,\ 0\lt x\le \pi ...
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2answers
80 views

“Every function can be represented as a Fourier series”?

It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series. So this got me thinking about the ...
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0answers
27 views

Differentiation Property of Fourier Transform

I've been asked to show that the Fourier Transform satisfies a list of properties, and I can show that the $m$-th derivative of a FT is multiplied by $(-i\xi)^m$ by inductively applying the original ...
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0answers
18 views

DFT of subdomain of periodic domain

$f(t_i,x_j)$ is a solution of stochastic differential equation on grid. $j=[0,N+1]$, $i=[0,\infty]$ and boundary conditions are periodic: $f(t_i,x_0) = f(t_i,x_N)$ and $f(t_i,x_{N+1}) = f(t_i,x_1)$ ...
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1answer
30 views

Fourier expansion of absolute value of a periodic function

For an arbitrary periodic function p(x), whose period and Fourier expansion might have been known in advance, how can we get the Fourier expansion/coefficients of |p(x)| from them? Or, if possible, ...
0
votes
1answer
30 views

$f$ is smooth and periodic function,$\exists \lambda$ such that $f^{(4)}=\lambda f$ prove:$\exists \lambda$ such that $\lambda= (\frac{2\pi n}{T})^4$

Given $f:\mathbb{R} \rightarrow \mathbb{C}$ is smooth and periodic with period $T>0$ and exists $\lambda \in \mathbb{C}$ such that $f^{(4)}(x)=\lambda f(x)$ for any $x \in \mathbb{R}$. prove: ...
4
votes
1answer
96 views

Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
2
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1answer
52 views

Identity of $\coth $ using Fourier series

The exercise wants me to prove the identity $$\pi \coth \pi a= \frac{1}{a}+ \sum_{n=1}^{\infty}\frac{2a}{n^2+a^2}$$ using the Fourier series of $\cosh ax, \; x \in [-\pi, \pi], \; a \neq 0$. ...
1
vote
1answer
43 views

Can a sine wave be expressed as a sum of square waves.

The opposite is possible, throught simple fourier analysis. For this question, suppose we have the periodic functions square wave functions: $f(x) = \begin{cases} 1 & 0\leq x < 1 \\-1 & ...