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

2
votes
3answers
56 views

Writing a matrix as a linear combination of basis matrices…

BACKGROUND: I have recently found (probably well known, but I had never seen this before) that a matrix can be written as a linear combination of the outer products of its eigenvectors where the ...
-4
votes
0answers
15 views

how can i applay fourier transform to a fourier series+reves+fillter respons

hellow and thenks for helping. i got a tesk in class. we should design a low-pass filter (butter worth fillter) to a given band and stop band freq. i need to cunculate the respont of a periodic rect ...
1
vote
1answer
48 views

Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - ...
0
votes
0answers
16 views

Higher Order Fourier Transform

How do you extend Fourier transforms beyond the 1-d case? I just learned about the transform and I am curious if anyone has some good information I can study over the summer.
0
votes
0answers
26 views

Convergence of a Fourier series on the unit circle

I have a complex-valued function defined as $$\psi(z) = \sum_{j\in\Bbb Z} \psi_jz^j$$ We of course know that $\sum_j\lvert\psi_j\rvert < \infty$ implies $\psi(z)$ is well-defined (finite) on the ...
0
votes
0answers
11 views

Length of a line approximated by Fourier series

I was recently solving some simple exercises where you approximate a square wave with a Fourier series. As you add more and more terms to the Fourier series, the function becomes close in shape to ...
0
votes
0answers
23 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ ...
2
votes
0answers
57 views

Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$

I'm sure easier proofs exist, but I have to specifically use the method in the picture: This is what my attempt is: First, I did some manipulation to figure out that $$ \sum_{k=1}^\infty \frac ...
0
votes
1answer
19 views

Closed form of integral $\int_a^b e^{-ix^2} dx$

Does any know how to find the closed form of integral $\int_a^b e^{-ix^2} dx$ for any real $a$ and $b$. It seems that I need to use the fresnel integrals.
0
votes
1answer
20 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) ...
2
votes
0answers
79 views

What is the name of that theorem?

Here is the statement : Let $f:\mathbb{R}\to \mathbb{C}$ a continuous map which is $\mathcal{C}^1$ by pieces and such that $f\in \mathcal{L}^1(\mathbb{R})$. Moreover, $\hat f \equiv 0$ in ...
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 ...
0
votes
1answer
28 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
60 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
23 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
12 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
14 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. ...
0
votes
1answer
25 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
35 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
12 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
22 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
21 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
19 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
113 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
25 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
26 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
20 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 $$\Delta f(x,y) = \frac{\partial}{\partial x^2} f + \frac{\partial}{\partial y^2}f$$ ...
0
votes
0answers
29 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
20 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 $\Bbb R$. ...
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
40 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 ...
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 $$ ...
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
31 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
74 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
34 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 ...
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
62 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 ...
0
votes
1answer
65 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
22 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 ...
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 ...