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.

learn more… | top users | synonyms

1
vote
0answers
18 views

Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that $...
1
vote
1answer
30 views

$x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$

I am trying to establish the following $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$ The right sight looks the the Fourier expansion of ...
1
vote
1answer
66 views

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert ...
0
votes
0answers
26 views

Fourier series for discontinuous function

I am a bit confused with the Fourier series. The first step should be to determine if my function is odd or even, then find the coefficients (with eventually the shortcut for odd or even function) and ...
0
votes
1answer
13 views

How would it looks the Fourier series of this signal?

This is a kind of digital signal I'd like to re-create. i.e. I'd like to get N samples that will describe this signal: even better if it satisfy the Nyquist theorem (thus, sample-rate is 2x ...
0
votes
0answers
15 views

Complex exponential fourier series

Given $$ x(t) = \sum_{-\infty}^{\infty}\frac{1}{T_0}(t-nT_0)(-1)^n[u(t-nT_0)-u(t-(n+1)T_0)]$$ where $n\in \mathbb{Z}$, $T_0$ is the period, and $u(t)$ is the unit step function, sketch $x(t)$ and ...
-1
votes
1answer
26 views

Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually ...
0
votes
0answers
16 views

Poissions Equation (Laplace)

$$\begin{align} u''_{xx}&+u_{yy}= x, \quad 0<x<1, \quad 0<y<1,\\ \\ u(x,0)&=u(x,1) = 0, \\ u(0,y)&=u(1,y) = 0,\\ \end{align}$$ Having some problems with Poissons Equation. I'...
0
votes
1answer
28 views

Fourier Series of a sum of two functions [closed]

Is the Fourier series of a sum of two functions $f,g$ the term by term sum of the Fourier Series?
1
vote
1answer
36 views

Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
0
votes
0answers
13 views

For fourier series g(x), prove that the fourier series for the integral G(x) can be found by term-by-term integration of g(x)

I want to prove that if I have a fourier series of the form $g(x) = a_0/2 + {\sum_i}^\infty a_icos(ix) + b_isin(ix) $, the fourier series of G(x) $-x*a_0/2$ can be found by simply integrating g(x) ...
1
vote
0answers
34 views

Why can we calculate the Fourier series of $x^2$ in any interval $[-l,+l]$?

We know that a function must satisfy Dirichlet's Conditions before it can be expanded in Fourier series. And Dirichlet's Conditions strictly require a function to be periodic in the interval in which ...
1
vote
1answer
27 views

How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
0
votes
2answers
22 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
2
votes
1answer
21 views

Combination of even and odd functions

Can someone help me how to show that any function $f(x)$ defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by "...
1
vote
2answers
114 views

Evaluating infinite series $\sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2}$

I have no idea to approach this problem. Mathematica gave the sum to be $$ \sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2} = \frac{\pi}{4a} \tanh(\frac{a \pi}{2}) $$ How can I analyze this?
2
votes
1answer
28 views

Fourier series on incomplete data [closed]

Given a periodic function that's only partly specified, e.g.: $$f(\theta)=\begin{cases}1 & \text{if } \cos(\theta)>a\\ -1 & \text{if } \cos(\theta)<-a\end{cases}$$ Obviously the ...
0
votes
0answers
27 views

Given Fourier coefficients of a function , find the function

Given these Fourier coefficients: $$ X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases} $$ I want to find the analytical expression for the function. What i tried was ...
1
vote
0answers
39 views

Is there a general rule to find period of multiplied functions?

We know that $g(x)$ and $f(x)$ are both periodic and trigonometric functions and we also know its period interval. How can we find the period of the function $f(x)g(x)$?
0
votes
1answer
21 views

Suppose $f(x,y)$ has double Fourier series, find Fourier series of $\Delta f$

Suppose $f(x,y)$ has double Fourier series $\sum a_{n1n2} e^{in_1 x} e^{in_2 y}$. Then I have $$\Delta f(x,y) = \frac{\partial}{\partial x^2} f + \frac{\partial}{\partial y^2}f$$ $$=\frac{\partial}{\...
0
votes
0answers
30 views

Dirac function expansion

In my book it is said that Dirac function $\delta(\tau)$ can be expanded as: $$ \delta(\tau)=(\beta \hbar)^{-1}\sum_{n \in even} e^{-i\omega_n\tau} $$ where $\omega_n=\frac{n\pi}{\beta\hbar}$, and $\...
0
votes
0answers
26 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (...
1
vote
0answers
21 views

Uniform convergence of Fourier series given certain conditions

If $f$ is a continuous, $a$-periodic and piecewise differentiable function on $[0,a]$ with piecewise continuous derivative on $[0,a]$, then $(f_N)$ converges uniformly to $f$ over $\Bbb R$. With: $...
0
votes
1answer
11 views

Coefficients of Fourier Series of (Cos(t))^3

I have to do the problem through the Sine/Cosine formulation of Fourier Series, so I'm talking about those coefficients. The interval is [-π, π]. I did the problem and checked it via Wolfram ...
0
votes
1answer
42 views

Fourier series coefficients in PDEs

I have a problem that involves solving a PDE using separation of variables. For context, here is the question: $u(x,t)$ is the displacement of a string at position $x$ and time $t$, which is ...
1
vote
0answers
27 views

How do I calculate the Fourier Transform of this signal?

The Context: Find $X(ω)$ which is the frequency domain representations of $x(t)$. $$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$ This my professor's solution: As we can see, the ...
0
votes
0answers
19 views

Shifting the Fourier Series?

If $f(x)$ is some periodic function, I know how to express the shift $f(x-a)$ in the complex formulation of the Fourier series. However, I was wondering how such shifting affects the coefficients $a$o,...
1
vote
0answers
35 views

Use the Fourier Series of $f(x)=x^2+1$ to find the sum of the series

I have found the Fourier Series of $f\left(x\right)=x^{2}+1$ on the interval $\left[-\pi, \pi\right]$ extended periodically to $\mathbb{R}$ to be $$ f\left(x\right)=\dfrac{\pi^{2}}{3}+1+\sum^{\infty}_{...
1
vote
0answers
27 views

Fourier series in spherical coordinates?

I'm reading an article and he just state: let $f\left(\theta,\varphi\right)$ be of this form $$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},$$ I'm on the unitary ...
0
votes
1answer
32 views

How to prove that $f(x) = x(1-x)$ converges to a Fourier series?

The solution to an exercise I've done approximates $ f(x) = x(1-x)$ as a Fourier series, but does not mention how I can prove that $f(x)$ is indeed equal to the solution series. What I've done is : $...
5
votes
2answers
75 views

How to evaluate this series using fourier series?

With the help of Hermite's Integral,I got $$\sum_{n=1}^{\infty }\frac{1}{n}\int_{2\pi n}^{\infty }\frac{\sin x}{x}\mathrm{d}x=\pi-\frac{\pi}{2}\ln(2\pi)$$ I'd like to know can we solve this one using ...
0
votes
1answer
35 views

Determing an inverse Fourier transform

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to Q1: $...
0
votes
1answer
16 views

inverse fourier transform of w*e^w

I have the function \begin{align} F^{-1}\{{λe^{-|λ|}}\} \end{align} How can we find the inverse Fourier transform? The correct answer is: \begin{align} \frac{-2ix}{π(1+x^2)^2} \end{align} Can ...
2
votes
1answer
67 views

Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...
0
votes
1answer
16 views

Can we relax the hypothesis of Uniqueness theorem for Fourier series?

I know this fact: "Suppose that $f\in L^{1}(\mathbb T)$ and $\hat{f}(n)=0$ for all $n\in \mathbb Z,$ then $f=0 $ all most everywhere on $\mathbb T$." My Question is: Suppose that $f\in L^{1}(\...
0
votes
1answer
66 views

Fourier Series of a piecewise-linear function

One is asked to determine the Fourier series of the function $$ f(x)= \left\{\matrix{ 0 & \hbox{(for $-\pi\le x<0$)} \cr x & \hbox{(for $0\le x<\pi $)} }\right. $$ where $f(x+2\pi)$ = $...
0
votes
0answers
18 views

Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$ A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
1
vote
1answer
24 views

Index of a derivative operator on a circle

Let $D: C^{1}(S^{1}) \rightarrow C(S^{1})$ be an operator defined as $D(f)=f'$. I would like to find its index (on the road proving that it's a Fredholm operator). First, if $f \in ker(D)$, then $f$...
0
votes
0answers
13 views

How to calculate Fourier coefficient of $f\in C^{\infty} (\mathrm{T^3})$?

I was trying to calculate the $k$-th fourier coefficient $c_k$ of some smooth functions on $T^3$, say $k=(m,n,p)\in \mathbb{Z}^3$. In a write-up I found online, it has the following definition: $$c_k =...
0
votes
0answers
19 views

What effect does sampling time have on a Fourier Series sum?

What effect would the sampling time of this Fourier sum have on it's accuracy? Is this to do with Nyquists theorem? or am I heading in the wrong direction with this question? Cheers
0
votes
0answers
33 views

Fourier sine and cosine series: reconstruction is shifted with respect to measured data

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
1
vote
0answers
24 views

Prerequisites for Fourier Series/Self-Study?

What would be the prerequisites for a typical Introduction to Fourier Series taught at a university. Is an introduction to a rigorous treatment of calculus typically expected or not? So far I've ...
1
vote
0answers
30 views

How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
0
votes
0answers
13 views

Partial Fourier Series Error

I calculated the coefficient of the complex fourier series of a trapezoidal signal to be: $$c_n = \frac{\tau}{T} \frac{sin(0.5n\omega_0\tau) \cdot sin(0.5n\omega_0\tau_r)}{(0.5n\omega_0\tau) \cdot (0....
2
votes
0answers
23 views

Bounding Fourier coefficients using translates of a function

Having real trouble solving this problem, I know there's probably something obvious that I'm missing but it's driving me mad! Exercise 1.3. Let $f \in C(\mathbb{R} / \mathbb{Z})$. For $y > 0$ ...
0
votes
1answer
21 views

Is square wave is an odd function if there is a dc shift?

Is periodic square wave is an odd function if there is a dc shift?(a0 != 0) And how do I determine whether square wave is odd or not when there is a dc shift.
0
votes
0answers
35 views

Computing the complex fourier series for triangular wave from trigonometric coefficients

I'm trying to figure out how to compute the complex Fourier series for the triangular wave, given the trigonometric coefficients. The book gives as a result for the complex series the following: $$ f(...
0
votes
0answers
17 views

Heat and Wave equation - Green's function versus Fourier series?

I am learning how to solve the heat and wave equation in bounded domains in 1 and 2D as well as in $\mathbb{R}$ and $\mathbb{R}^2$. In the latter case I have learned the representation formulas i.e. ...
2
votes
2answers
101 views

Why does specifying an interval for a function make the function odd or even?

I am currently reading about Fourier series and Orthogonality of functions and Complete Sets of functions. Below are two extracts from the book I'm reading for which I simply do not understand: <...
0
votes
1answer
24 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) $\omega^2=...