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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How do I calculate the Trigonometric Fourier Series Coefficients of the following function?

I'm having trouble figuring out how to find the Trigonometric Fourier Series of the following function: $${e^{t+1}+e^{j(2t+3)}}$$ I know the following: The Trigonometric Fourier Series is defined ...
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20 views

Fourier series of x

I'm trying to find the first order Fourier series of $f(x) = x$ using the formula : $f(x) =\sum_0^{\infty} (c_n e^{in\omega x} + c_{-n}e^{-in\omega x})$ with: period $T = 2\pi$ order $n = 1$ ...
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58 views

Proof: $f$ square-integrable $\Rightarrow f$ absolutely integrable on $[0, 2\pi]$

In a book I found the following statement: Let $\varphi(x)$ and $\psi(x)$ be square integrable, then $|\varphi \psi| \leq \frac{1}{2} |\varphi^2 + \psi^2|$. This implies, that every square ...
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1answer
60 views

If a continuous function on $[0,\pi]$ integrates to zero against cosines, it is identically constant

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
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1answer
26 views

fourier series for g(x)=x between -pi and pi

Consider the following function defined on a finite interval: $$g(x) = x, 0\leq x\leq \pi $$ (3) (a) Sketch an even periodic extension of g(x). (b) Show that the Fourier cosine series representation ...
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2answers
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Fourier series, instantly determining $b_n$ once $a_n$ is found.

Find the Fourier series of the following function: $f(x) = \left\{\begin{align} 1+x,\quad -1\lt x \lt 0 \\ 1-x,\;\;\;\quad 0\lt x \lt 1\end{align} \right.$ $f(x+2) = f(x),\quad\quad -\infty \lt x ...
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1answer
53 views

How did Fourier series lead to the development of rigorous analysis?

Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him ...
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1answer
23 views

Fourier Series Coefficient Question

In calculating the Fourier Coefficients a0, an, bn: Why are the an and bn coefficients integrated over 2 times the inverse of the period, 2(1/T) while the a0 coefficient is integrated only over one ...
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28 views

Showing a series is not the fourier series of a riemann integrable function.

I want to show that the series $\sum_1^\infty \frac{sin(nx)}{\sqrt{n}}$ is not the Fourier series of a Riemann integrable function on $[-\pi,\pi]$. I was going to do this by showing that the partial ...
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22 views

Fourier series on an arbitrary interval

Let $f(x)=x$ on $[0,\pi]$. I'm stuck trying to find the Fourier series on that interval. $a_0=\frac{1}{\pi}\int\limits_{0}^{\pi}f(x)=\frac{\pi}{2}.$ ...
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1answer
28 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
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1answer
51 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
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1answer
34 views

Fejer's theorem with Riemann integrable function

If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then $$ \lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = ...
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1answer
30 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 ...
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36 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)= ...
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15 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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2answers
39 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 ...
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1answer
29 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 ...
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1answer
42 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) ...
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1answer
28 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 ...
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76 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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44 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)\\ $ ...
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24 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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1answer
33 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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119 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 = ...
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32 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 ...
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1answer
22 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: ...
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35 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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1answer
23 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 ...
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55 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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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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39 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,\\ ...
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45 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 ...
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228 views

problem on a function being identically zero

Let $f:[0,\pi] \to \mathbb{R}$ be a continuous function such that $f(0)=0$. If $$ \int_0^\pi f(t)\cos nt\, dt = 0 $$ for all $n \in \mathbb{N} \cup \{0 \}$, is $f$ identically zero?
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35 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 ...
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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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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 ...
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1answer
32 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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29 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 ...
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103 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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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 & ...
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1answer
22 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 $\int_{0}^{L} f_e(x) dx$ ...
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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 ...
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1answer
163 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 ...
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5answers
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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 ...
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23 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 ...
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1answer
42 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 ...
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1answer
38 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 ...
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35 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{ ...
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61 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 ...