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.

learn more… | top users | synonyms

0
votes
0answers
12 views

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 ...
0
votes
0answers
15 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 $n = 1$ ...
0
votes
0answers
30 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 ...
0
votes
1answer
21 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
votes
2answers
30 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 ...
1
vote
1answer
40 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
votes
1answer
11 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 ...
1
vote
0answers
17 views

Fourier transform and conjugate variables

When you make a Fouriertransform of a function of time $f(t)$, it is said that it's Fouriertransform is a function of frequency $\widetilde{f}(\omega)$. The same argument goes for position and ...
1
vote
0answers
23 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 ...
0
votes
0answers
11 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. $a_0=\frac{1}{\pi}\int\limits_{0}^{\pi}f(x)=\frac{\pi}{2}.$ ...
0
votes
0answers
13 views

Determine Periodicity with imaginary numbers

I have a function $\cos(a\cdot n) + j\sin(a\cdot n)$, for $|n| < 10$ ($a$ is some constant). My textbook says that this function is not periodic; I was wondering why this is? Is it because n is ...
1
vote
1answer
26 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 ...
2
votes
1answer
35 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, ...
1
vote
1answer
27 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} } = ...
0
votes
0answers
19 views

Complex Fourier coefficients and series

I need help trying to find the complex Fourier coefficients for the functions $\cos(3x)$ $\sin(2x)$ I know the equation for finding the coefficients and how to plug it in but I'm confused in how ...
0
votes
1answer
24 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
votes
1answer
28 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)= ...
1
vote
0answers
19 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 ...
0
votes
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 ...
3
votes
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 ...
1
vote
1answer
17 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
vote
1answer
38 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) ...
0
votes
1answer
21 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 ...
1
vote
0answers
69 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 ...
0
votes
1answer
38 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
votes
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 ...
0
votes
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 ...
0
votes
1answer
35 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 = ...
0
votes
0answers
22 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 ...
1
vote
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: ...
2
votes
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= ...
0
votes
1answer
18 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 ...
-1
votes
0answers
16 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 ...
0
votes
1answer
39 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
votes
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 ...
0
votes
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
vote
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
votes
0answers
26 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
votes
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. ...
1
vote
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 ...
1
vote
1answer
22 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 ...
0
votes
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 ...
3
votes
1answer
94 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 ...
0
votes
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
votes
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
votes
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 $\int_{0}^{L} f_e(x) dx$ ...
0
votes
0answers
20 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
votes
1answer
24 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 ...
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
vote
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 ...