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
690 views

fourier series of absolute value of function

I am trying to find the Fourier series of $$ |\cos(x)| \text{ from } -\pi \leq x<\pi$$ I know that the $$ b_n $$ terms go to 0 because we have the integrand as an odd function of x. But how can ...
2
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1answer
24 views

Determining if two expressions are equal, in order to ensure a Fourier series is correct

Motivation: I have a question that asked me to find the Fourier series of some function $f(x) = \left\{\begin{array}A,\quad -1\lt x \leq 0 \\ Ax, \quad 0 \lt x \leq 1 \end{array}\right.$ periodic on $...
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0answers
33 views

Basis for quaternionic functions

We know that the set of functions $\{1,\cos x, \sin x, \cos 2x, \sin 2x, ... \; | \,x \in \mathbb{R} \}$ is a basis in the space $L^2_\mathbb{R}[-\pi,\pi]$ . Given a quaternion $z \in \mathbb{H}$ ...
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2answers
63 views

Showing two things are equal by Fourier series

Given the Fourier series for the function: $$f(x) = x+\frac14x^2 \quad -\pi\leq x \lt \pi$$ $$f(x)=f(x+2\pi) \quad -\infty \leq x \lt \infty$$ is $$\frac{\pi^2}{12}+\sum \limits_{n=1}^\infty (-1)^n \...
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0answers
34 views

How to solve an inhomogenous PDE using Fourier Transform

$u_{tt}=u_{xx}+(8-64x^2)e^{-4x^2}$ $u(x,0)=e^{-4x^2},u_t(t,0)=0$ $0<t<\infty,-\infty<x<\infty$ By Fourier Transform $\frac{d^2u(w,t)}{dt^2}=F[u_{xx}]+F[(8-64x^2)e^{-4x^2}]=-w^...
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0answers
46 views

A question on convergence of derivative of power series

This is a question from Fourier Analysis with Applications by Folland. First we write Fourier series for $$e^{\theta}=\sum c_ne^{in\theta}$$ We differentiate this series term by term to obtain $$e^{\...
1
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1answer
209 views

Fourier Series of Real-valued Functions

Context: For a $2\pi$-periodic bounded function $f:\mathbb{R}\to\mathbb{C}$, we define the complex Fourier coefficients of $f$ by $$ \hat{f_k}:=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}\,dx. $$ We call ...
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0answers
53 views

Fourier Cosine series expansion for two dimensional function

I have a two dimensional function with its values and range. I need to expand the function in Fourier cosine series. The function as follows: $$f(x,y) = \begin{cases} A &, -\frac{L}{2} + 2nL < ...
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3answers
77 views

Fourier series of $f(x)=1$

$\displaystyle f(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty a_{n}\cos nx$, where $a_{n}=\frac{2}{\pi}\int_0^\pi f(t)\cos(nt) \ dt$, if $f$ is even. But for $f(x)=1$, the left side goes to $0$. How can I ...
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2answers
31 views

Using Weistrass Approximation Theorem to define fourier series convergence.

Weistrass Approximation Theorem: Let f be continuous on [-$\pi$,$\pi$] with $f(-\pi)=f(\pi)$. Then for each $\epsilon>0$ there is a trigonometric polynomial T such that $|f(x)-T(x)|<\epsilon$ ...
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0answers
58 views

Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on $[0,\pi]$...
2
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1answer
68 views

Express $f(x)=\sin{x}$ as an even function

Express $f(x)=\sin{(x)}$, with $(0 < x< \pi )$ as an even function, $f(x+ 2\pi)=f(x)$ The topic is on Fourier Series. I have the following so far: Since $f(x)$ must be an even function, we ...
15
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1answer
328 views

Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi nx)\right\}}$$...
0
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1answer
69 views

Fourier series of $f(x) = x - [x]$, where $[x] = n \in Z$ s.t. $n \leq x < n+1$

How do you find the Fourier series of $f(x) = x - [x]$, where $[x] = n \in Z$ s.t. $n \leq x < n+1$? I am familiar with Fourier series and use the following definition to solve them: $f(x) = \...
1
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1answer
33 views

How to write a fourier series using periodic boundary conditions

Would writing $$ f(x) = x^2 $$ as a Fourier series using periodic boundary conditions on $-L < x < L$ with a basis of $$ e^{\frac{i\pi nx}{L}} $$ be just \begin{align}\bigl\langle e^{\frac{i\...
1
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1answer
142 views

Solving $\sum_{n=1}^{\infty} \frac{1}{n^2}$ using the fourier series.

Please do NOT solve the problem, I just need some help, not a full solution. I would like to try this myself. Find $\zeta(2) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$ The fourier series for ...
2
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1answer
142 views

Questions about Fourier Series

I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ ...
3
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1answer
72 views

Did I calculate this Fourier series correctly?

If we use the definition of the Fourier series in the following way: $$f(x) = \frac{A_{0}}{2} + \sum_{n=1}^\infty A_n \cos(nx) + B_n \sin(nx)$$ then if $-\pi < x < \pi$, and $f(x) = \...
0
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1answer
68 views

Helpful Integrals for evaluating Fourier series, my book is wrong?

I don't understand why my book is claiming the following for any $n$ or $w_0$ this is always the case over one period. I think it depends on the $w_o$ really. I have proof too, but I just want another ...
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2answers
27 views

Help with setting up the Fourier series for the following functions.

i. $f(x) = \operatorname{sgn}(x)$ for $-\pi < x < \pi$ where $$\operatorname{sgn}(x) = \begin{cases} 1, & x>0, \\ 0, & x=0, \\ -1, & x<0. \end{cases} $$ ii. $f(x) = \...
3
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0answers
42 views

Discrete Fourier Transform of a shift of a tuple over a finite field

Let $a = a_0 a_1 \cdots a_{N-1}$ be a sequence over a finite field $\mathbb{F}_q$, where $N \mid q^n-1$ for some $n$. Let $\xi_N$ be a primitive $N$-th root of unity in the extension $\mathbb{F}_{q^n}$...
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0answers
57 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
3
votes
2answers
208 views

Different Versions of Fourier Series? What about Uniqueness?

Let $f(x)$ be a function, then for its Fourier series $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) $$ I found two different definitions (both yielding different series)....
0
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1answer
28 views

write y(x) according to sin(d+(ay+b)/cx)=y

I end up with the formula $\sin (d+\frac{(ay+b)}{cx})=y $, and try to write $y$ as a function of $x$. There can be multiple solutions (of $y)$ to $\sin(d+\frac{(ay+b)}{cx})=y$ pretending $x$ is known, ...
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0answers
208 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
85 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 a ...
0
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1answer
35 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 ...
2
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2answers
47 views

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 \...
2
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1answer
112 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 ...
0
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1answer
37 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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1answer
309 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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1answer
44 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 ...
3
votes
1answer
273 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
44 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} } = \frac{1}{2}...
0
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1answer
32 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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1answer
54 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)= \left\{\begin{...
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2answers
49 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
193 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
49 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
60 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 x}{L}\...
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0answers
81 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
70 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
38 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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vote
1answer
174 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
5k 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 = \frac{\sin(k\...
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1answer
39 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: ...
2
votes
1answer
44 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= -\infty}^{\...
0
votes
1answer
31 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 \...
2
votes
2answers
75 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 \,...
0
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2answers
40 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,\\ {\...