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
7 views

Non periodic Fourier Series Point Convergence

If $f$ is a real-valued non-periodic continuous function that is differentiable at the point $x_0$, is it true that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
0
votes
0answers
22 views

Heat Equation of Very Long Bar [on hold]

I need to find the solution in integral form with initial condition $u(x,0)=f(x)$ where $f(x)=1$ if $\mathrm{abs}(x)<a$ and $0$ otherwise. Any help is greatly appreciated!
0
votes
1answer
20 views

Existence of Solutions to PDEs - How do I know I've got them all?

I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't ...
1
vote
2answers
26 views

Limit in combination with an infinite series

How would I go about showing the following limits that involve infinite series $$ \lim_{x \to 0^{+}} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2k+1}} \sin (2\pi n(x - \frac{1}{2})) = 0 \text{ with } k \in ...
0
votes
0answers
15 views

Sturm-Liouville eigenfunction expansion converges to 1-x instead of -1

I am presented with a Sturm-Liouville problem $y''+\lambda y = 0$, $y(0)=0$, $y'(1)=0$. I solve this to obtain eigenfunctions of the form $y = \sin{(n-\frac{1}{2})\pi x}$. Now I wish to find a ...
2
votes
2answers
25 views

Help establishing a bound on the Fourier coefficients of a bounded $2\pi$ periodic function that is discontinous at the end points?

This is from a practice midterm, and I'm having trouble with the first part. Suppose $f$ is a $2\pi$-periodic function that is continuous and differentiable on the interval $[-\pi, \pi]$, but has jump ...
4
votes
1answer
18 views

Fourier Series for a conformal map on unit disk

Given that a conformal map on the disk $\mathbb{D}$ will always have the form $f(z)=\lambda \displaystyle\frac{z-w}{1-\overline{w}z}$ for some $\lambda\in \partial \mathbb{D}$ and some $w\in ...
0
votes
0answers
15 views

Fourier Expansion of Hill's lunar problem

all! For my class I have to expand the following equation $y''(x)=4(\omega^2+q(x))y(x)$ in Fourier coefficients $y(x)=\frac{1}{2}y_0 + \sum^\infty_{n=1}y_n \cos(2nx)$ $q(x)=2\sum^\infty_{n=1}t_n ...
0
votes
0answers
15 views

How this result in archived in Fourier series

I was reading some notes about functions of symmetry in Fourier series and came across the following result for a function with symmetry of an odd quarter wave $$\begin{align} ...
3
votes
0answers
61 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
2
votes
0answers
7 views

half range fourier series, even and odd extension

Hello, I have some problems understanding what is above on the image. Firstly, he defines an "odd extension" of any function. I don't really understand what this means, how is it an "odd extension" ...
0
votes
1answer
16 views

Fourier series coefficient miscaluculation

In a nice introductory paper about Bernoulli numbers that I found, the following claim is made (p. 5, theorem 4.3) The Fourier series of $x$ is given by $b_n = \dots$ (not important, it is wrong in ...
2
votes
0answers
26 views

Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
2
votes
2answers
32 views

Fourier series of complex diff eq

Can I just use Euler's identity to construct the Fourier Series since it is complex? I was personally thinking I could, but I wanted to be doubly sure.
0
votes
0answers
26 views

Please help me with Fourier series problem!

(a) Find Fourier series of $f(x)$ on $[-L,L]$ $$f(x)=\begin{cases} x(L-x) & 0\le x<L \\ x(L+x) & -L < x < 0 \end{cases} $$ (b) Find $f'(x)$ and $\int_{-L}^x f(x)\, dx$ and the ...
0
votes
0answers
15 views

Bibliographic reference for $\sum_{n\in\mathbb Z}(z-n)^{-k}$

I am currently writing a paper which requires the closed-form expression of \begin{equation} S_k(z)=\sum_{n\in\mathbb Z}\left(z-n\right)^{-k} \end{equation} I believe $S_2$ is extremely classical ...
2
votes
2answers
69 views

Fourier Decompositon

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
1
vote
1answer
16 views

Determine the Fourier transform of $f(x) &

f(x)=1 if |x| < a or f(x) = 0 if |x| > a We use the formula $$ {1\over 2\pi} \int_{\infty}^\infty f(\bar x)e^{i\omega \bar x} $$ So is $f(\bar x)$ the same as $f(x)$ ?? In an answer they ...
0
votes
1answer
28 views

Fourier Series Convergence

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function that is differentiable at the point $x_0$. Prove that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
0
votes
2answers
65 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...
1
vote
1answer
20 views

Fourier series phasor form and sin/cos form

can anyone give me a link on how to convert the forms (from phasor to sine/cos and vice versa)? I am new to this and I can't find the convertion table with a valid explaination.
1
vote
1answer
24 views

Parseval's identity does not hold for constucted basis

As part of an exercise, I was asked to show that given an orthonoraml basis $(\varphi_1,\varphi_2,\varphi_3,...)$ in $L_2[-\pi,\pi]$, we can construct an orthonormal basis $(\psi_1,\psi_2,\psi_3,...)$ ...
0
votes
0answers
23 views

how manipulating the coefficients vector effects on the result of DFT?

given: calculate: note that the given DFT is from n order and we want to compute DFT's from 2n order. edit: this is my try of B. i don't see where the given DFT is used and how to proceed: ...
1
vote
0answers
16 views

Complex exponential argument to a function

In many texts on signal processing, the following notation is used to describe the Fourier transform of a discrete time signal $x$: $$ \hat{X}\left(e^{j\omega}\right) = ...
0
votes
0answers
12 views

Fourier Series of Complex Valued Functions

Write the Fourier series of functions in the space of complex valued functions $L^{2}[0,1]$, which we view as periodic functions on $\mathbb{R}$. Specify the coefficients of the expansion and also ...
0
votes
1answer
31 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
0
votes
0answers
21 views

Numeric Evaluation of Double Surface Integral over Greens Function with Singular Points

I'm currently using python to numerically evaluate the follow expression ...
0
votes
1answer
13 views

Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
1
vote
1answer
30 views

Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
0
votes
0answers
24 views

Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
2
votes
2answers
31 views

Getting fourier coefficient by integrating over half the period?

In the book Schaum's Outlines of Analog and Digital Communications solved problem 1.2, the author calculates the fourier coeffecient $C_0$ for the rectangular pulse train: where $a$ is assumed to be ...
1
vote
1answer
15 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
0
votes
1answer
37 views

Difficult integration

In my notes the lecturer takes the Fourier transform in $x, y$ and $t$ of $\phi(x,y,z,t)$ as: $$ \int_{-\infty}^{\infty}dt\, e^{i\omega ...
0
votes
0answers
12 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
1
vote
2answers
83 views

Is Fourier transform still writing a function as a series of sines and cosines?

In the Fourier series we write a function as a series of sines and cosines. Fourier transform seems to me to be totally different, we are not finding a series but rather a function $\hat f(w)$. So ...
3
votes
1answer
23 views

Fourier series and evaluation of another series

I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula: ...
1
vote
1answer
20 views

What kind of information is available in a Fourier series expansion of an analytic function that is not (readily) available in a Taylor series?

What kind of information is available in a Fourier series expansion of a real analytic function that is not (readily) available in a power series? When would one know to work with one over the other?
1
vote
0answers
35 views

Help for solving limi of the Complex Fourier Series

I need help for this exercise. Let: $ f:\left[ -T /2, T/2 \right]\rightarrow \mathbb{R}. $ I need show that $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
1
vote
2answers
111 views

Coefficient calculation on Fourier series !? [closed]

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
1
vote
0answers
18 views

Non Riemann summable Fourier series but Abel summable

A Riemann summable Fourier series is also Abel summable. I am looking for an example of a non-trivial Fourier series that is Abel summable at a point but NOT Riemann summable at the same point. Such ...
0
votes
1answer
18 views

Rectangular Width Fourier Function

Working on #7, I've tried writing out the Fourier transformation and plugging it into the formula and multiplying it with Wf, but I'm getting mixed up about how I'm allowed to combine integrals and ...
1
vote
0answers
26 views

Consider fourier transformations of $|p(\mathbf{r})|^2$

If we have $\mathbf{k}=(k_x,k_y,k_z)=\frac{2\pi}{L}(j,s,l)$ with $j,s,l \in \mathbb{Z}$ and we have $$p(\mathbf{r})=\sum_{\mathbf{k}}\tilde{p}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}} \implies ...
1
vote
2answers
31 views

Complex Fourier coefficients for $e^{|x|}$

I'm new to Fourier expansions and transforms, and I'm not sure how to proceed with this question. I know a function f(x) can be expressed as an infinite sum of $c_ne^{in \pi x/L}$, and that $c_n = ...
0
votes
0answers
10 views

Trying to find the Fourier series of $f(x)$, where $f(x)$ is a piecewise function that includes $E\;sin(\omega\;t)$.

Here's the full function I'm trying to find the Fourier series to: $$f(x) = \left\{ \begin{array}{lr} 0 & : -\frac{\pi}{\omega}\leq t\lt 0 \\ E\;sin(\omega t) & : 0\leq ...
0
votes
0answers
16 views

Show that the convolution of the two time domain functions satisfy the relationship Y(q) = H(q) * U(q).

The convolution of two time domain functions h(t) and u(t) is given by $$ y(t) = \int_{-\infty}^{\infty} h(t- \tau)u(\tau)d\tau $$ Show that the Fourier Transforms Y(q), H(q) and U(q) satisfy the ...
4
votes
1answer
61 views

Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant. Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< ...
1
vote
1answer
39 views

$\sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ does not converge as $\theta \rightarrow 0?$

We know that the series $H(\theta) := \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ is convergent for every $\theta \in (0,1)$ and for $\theta = 0$ the series tends to $+ \infty$. Is it ...
0
votes
0answers
15 views

A linear response system with a periodic input

I'm currently trying to solve the following exercise: A linear system is driven by a periodic input $f(t)$ such that $f(t+T)=f(t)$. The response $g(t)$ of the system is such that a sinusoidal ...
1
vote
1answer
31 views

Hard Integral [Heat Equation + Fourier Sine Series]

I encountered this integral while doing a heat equation problem in Advanced Calculus. How does the person evaluate the integral involving $$\int_0^\pi \sin x \cos (nx) \: dx $$ Can someone ...
-3
votes
1answer
22 views

Fourier transform of a scaled variable [duplicate]

If $f\hat(k)$ is the fourier transform of $f(x)$, what is the fourier transform of $f(x/c)$ where $c$ is a real number greater than $0$?