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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3
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2answers
33 views

Theorem of the convergence of the series of fourier! [duplicate]

During the demonstration of the theorem of the convergence of the series of fourier, my teacher wrote :$$ \frac{1}{2}+ \sum_{k=1}^{n} \cos(ky)=\frac{\sin((n+\frac{1}{2})y)}{2\sin(\frac{y}{2})} $$ he ...
0
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1answer
16 views

Express as a complex Fourier series

My function is $f(x)= \dfrac{1}{1-2e^{ix}} + \dfrac{1}{1-2e^{-ix}} $, which has been periodically extended by $2\pi$. I found $C_0$ to be $\pi$. I'm having trouble expressing $C_n$. All I have is ...
0
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0answers
17 views

Is a sine wave plus the sum of its odd harmonics symmetrical around the x axis at half the period of the fundamental?

I have a function such that $$x(t)=A_1 \sin(2 \pi f t+\phi_1)+A_2 \sin(2 \pi (3f) t+\phi_2)+...+ A_n \sin(2 \pi ((2n+1)f) t+\phi_n)$$ Is such a function symmetric around the point that is half ...
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0answers
19 views

Setting up my Fourier series for $B_n$

Related but not necessary to know: here Looking at the temperature distribution in an infinitely long cylinder of metal with insulated sides and initial temperature distribution $f(x)= ...
0
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0answers
6 views

Relation between the fourier series coefficients of $x(t)$ and $x(at+b)$

Consider function $x(t)$ is periodic with period $T_0$ and we call its fourier series coefficients: $a_k$ . Take $y(t) = x(at+b)$ and with fourier series coefficients $b_k$ . What is the relation ...
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1answer
16 views

Sufficient Condition for the convergence of Fourier Series

I'm studying real analysis and I know about derivative, Riemann integral, sequence and series, basic concepts. I'm having trouble understanding the sufficient conditions for a Fourier series of a ...
1
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1answer
34 views

Is it always the case that lower frequencies contribute the most in a Fourier series?

Is it always the case that lower frequencies contribute the most in a Fourier series? Or to put it in other words, in the equation: $$f(t)=a_0+\sum^\infty_{m=1} a_m\cos \left(\frac{2\pi mt}{T}\right) ...
1
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1answer
65 views

Complex Fourier Series. I Might Neeed Some Help On This Problem

The Problem: If $f(x) $ is a real funciton, rewrite the integral: $$ \frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} \, dx$$ in terms of the usual Fourier Coefficients, $A_n$ and $B_n$ The attempt: Recall ...
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1answer
20 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi ...
0
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1answer
26 views

How to plot fourier series in matlab

For homework (with no prior experience in matlab, guh.) I'm asked to do the following: Plot the (2N + 1)-term approximation $$\sum\limits_{k=-N}^N{a_ke^{jk\omega_0t}}$$ where $a_k = ...
2
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1answer
31 views

Can you help me with this Complex Fourier Series Problem?

Find the Complex Fourier Series of $F(x) = \cos(2x) + \sin(x)$ on the interval $[-\pi, \pi]$ Here is my attempt: The complex Fourier Series is in the form $\cos(2x) +\sin(x) = \sum_{n= ...
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0answers
154 views
+100

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
0
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1answer
32 views

Rewriting partial differential equation

I have some trouble rewriting a partial differential equation, more specifically the heat equation in one dimension: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t)\\ $ ...
0
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1answer
17 views

Fourier cosine series for a interval $[0, l]$

It is asked to find the Fourier Cosine Series for the function defined by $$f(x) = \cos \frac{\pi x}{l}, x \in [0, l/2]$$ $$f(x) = 0, (l/2, l]$$ I thought it should be $$\frac{a_o}{2} + \sum a_n ...
0
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1answer
23 views

I might need some help on this Complex Fourier Series Problem

Here is the problem: Use the Complex Fourier Series on $[-L,L] $ with complex coefficients to find a representation of $\frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} dx$ Here is my attempt: The ...
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0answers
20 views

I need help resolving my problem with DFT

I've been working to understand DFT and my results are not what I would expect. For clarity, I'm using C for T&E and my question isn't C related. My problem is in the DFT and my understanding of ...
0
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1answer
19 views

Can piecewise $C^{1}$ on $[a,b]$ imply Lipschitz continuity

I saw a statement that if $f$ is continuous,$2\pi$-periodic function which is $C^{1}$ piecewisely on $[-\pi,\pi]$, then its Fourier series converges uniformly to $f$ on $[-\pi,\pi]$. I was wondering ...
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1answer
17 views

Why is matlab giving me a single answer when I divide by a vector?

I'm attempting to do a stem plot of $\frac{sin(k2D\pi)}{k\pi}$ in matlab. Following is my procedure: ...
3
votes
1answer
93 views

Importance of groups $(\mathbb R,+)$ and $(\mathbb Z,+)$ for Fourier series

I have heard that the groups $(\mathbb R,+)$ and $(\mathbb Z,+)$ are the most important groups for Fourier series. Why is this the case? Supposedly, it has something to do with the fact that for any ...
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1answer
33 views

Nonseperable Hilbert space: Explicit ONB?

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
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0answers
14 views

If x(t) and X(w) are the Fourier Transform pair, then what makes pair with 1/X(w)?

If x(t) and X(w) are the Fourier Transform pair, then what makes pair with 1/X(w)? More elaborately, if the inverse Fourier Transform of X(w) is x(t) then, what is the inverse Fourier Transform of ...
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1answer
33 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
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0answers
13 views

Discrete Fourier vectors are the eigenvectors for any linear, constant coefficient, periodic, finite difference discretization on a uniform grid?

I came across the following statement: It can be shown that the DF vectors are always the complete set of eigenvectors of any linear, constant coefficient, periodic, finite difference discretization ...
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2answers
36 views

Terms in Fourier Series

Can any one explain why? $$\int_0^\pi \sin(nx)\sin(mx)\,dx=\begin{cases}0,&n\not=m,\\ {\pi\over 2},&n=m,\end{cases}$$ and $$\int_0^\pi \cos(nx)\cos(mx)\,dx=\begin{cases} 0, &n\not=m,\\ ...
1
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2answers
37 views

heat equation with fourier series

Original PDE $$T_t=\alpha T_{xx}$$ I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. Given ...
0
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0answers
23 views

Fourier Series Operation Is A Linear Operator

I am sort of stuck on this problem. Here it is: Show that the Fourier Series Operation is Linear, that is, show that the Fourier Series of $c_1f(x) +c_2g(x)$ is the sum of $c_1$ times the Fourier ...
2
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0answers
27 views

Can the Fourier Series be made “ Shorter ”?

I have tried to give only the intuitive part of my question and haven't included many specific details. Please help me frame it more precisely. I have inluded the symbol (*) where I need more details. ...
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1answer
20 views

Are fourier series of a periodic function expanded on different intervals equivalent

I was given an assignment by my instructor where i had to write the function $$ f(t) = \begin{cases} 1-t & 0\leq t < 1 \\ t-1 & 1 \leq t < 2 \end{cases}\\ f(t + 2) = f(t) $$ as a ...
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0answers
19 views

Fourier Series Problem

I am having trouble with this problem: For the following problem, sketch $f(x)$, the Fourier Series of $f(x)$, the Fourier Series and Fourier Cosine Series of $f(x)$. $f(x) = 1$. I think the ...
0
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0answers
20 views

Fourier series odd and even question

Is the use of odd and even functions for the Fourier series just so the formula for the fourier series becomes a shortcut for odd and even functions ? Does it mean that the idea of an extension can ...
0
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0answers
30 views

How to plot this function

How to plot this function in WolframAlpha or some other graphing calculator? $f(x) =\left\{\begin{matrix} 1 & -\dfrac{-2\pi}{3} \leq x \leq \dfrac{2\pi}{3}\\ -1 & ...
0
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1answer
22 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
0
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0answers
17 views

How can we solve a trigonometric system of equations with Fourier Transform?

Today the professor in linear algebra, when asked if there is any other way rather than linearize a trigonometric system of equations by letting variables as the trigonometric functions of each ...
0
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0answers
19 views

Bound on the integral of a differentiable function against sine functions

Consider a function $f: [0, L] \to \Bbb R$ and $k$ natural number. Suppose that $f', f'', ...$ $f^{n-1}$ are continuous and that $f^{k}$ is absolutely integrable. Show $$ \left| \int_{0}^{L}{f(x)\sin ...
0
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1answer
18 views

Fourier Series Proof

I have concerns about this problem Let $f_e(x)$ and $f_o(x)$ represent general continuous even and odd functions on $[-L,L]$. Prove that $\int_{-L}^{L} f_e(x) dx$ = 2 $\int_{0}^{L} f_e(x) dx$ ...
1
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1answer
21 views

Fourier coefficient one-period function

Define a function $f(x) =(2\cos(\pi x))^{10} $$f\in L^{1}$ so it's one-period. I would like to calculate the Fourier coefficient $\hat{f}(2)$. So we get $\displaystyle\hat{f}(n)=\int_{0}^{1}e^{-2\pi ...
2
votes
5answers
37 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
1
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0answers
51 views

Fourier Cosine Series question

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &2 \leq ...
0
votes
1answer
25 views

Fourier Sine Series and Cosine Series

This is the Fourier Series representation for a periodic function with period 2p, given in my lecture note. $\dfrac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(\dfrac{n\pi t}{p})+b_nsin(\dfrac{n\pi ...
0
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1answer
35 views

evaluate arithmetic sum by using fourier series

Hi I've been trying for 40 minutes to evaluate the sum of the following arithmetic series with no luck. $\sum_{n=1}^\infty \frac{sin(2k)}{k}$ I've tried to make this into a fourier series by ...
0
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0answers
31 views

Show that $x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$

Show that $$x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$$ for $ 0<x<\pi$ My idea: I've defined the periodic function $$f(x) = 0 \text{ if } x \in [- \pi, 0) \text{ ...
1
vote
0answers
26 views

Discrete Fourier Transform by hand

I have an assignment where I'm given the DFT of a sequence $x[n]$ as $X[k]=\{4,3,2,1,0,1,2,3\}$ and also $$y[n] = \left\{ \begin{array}[cc] xx[n/2] & \text{if n is even} \\ 0 & ...
0
votes
1answer
46 views

Calculating fourier series

I've a fourier series with a period = $2\pi$ that is even. f(t) = \begin{cases} 0 \text{, when: } 0<t<\pi-2 \\ \pi \text{, when: } \pi-2<t<\pi \end{cases} The functions trigonometric ...
5
votes
3answers
251 views

FFT with powers of 3

Classic Fast Fourier Transfrom (FFT) works fine, when $n$ is power of 2. How to generalize FFT procedure when $n$ is power of 3? Is it possible to easily modify the algorithm and preserve its ...
1
vote
1answer
67 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
0
votes
3answers
48 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
3
votes
2answers
46 views

Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
0
votes
1answer
47 views

Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
0
votes
2answers
28 views

Derivative of Fourier series

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ where $a$ and $b$ are lower and upper boundary, ...
2
votes
1answer
51 views

Using the Fourier Series of $f(t)=(t-\frac{1}{2})^{2}$ to deduce the sum $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$?

So this is a question in one of the previous tests: My approach (if you want just skip to step 3.):$$$$ 1. Formulation of the problem and calculating the constant term of the series $a_o$ I ...