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

2
votes
2answers
153 views

Sum of Fourier Series

I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by $$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$ in the interval $]-\pi,\pi[$ ...
2
votes
2answers
129 views

How to solve this equation by Fourier series?

$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$ Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
2
votes
3answers
182 views

Series help, fourier series

How do I know if a given function can be represented by a fourier series, that converges to the value of that function at non discontinuities. Also where did Fourier come up with the idea of ...
2
votes
3answers
447 views

About completeness of the Fourier series.

The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta + \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n ...
2
votes
1answer
5k views

Using Fourier series to calculate an infinite sum

Given the Fourier series of the $2\pi$-periodic function defined for $$-\pi\leqslant x \leqslant \pi$$ by $$f(x) = |x|$$ is $$ \frac{\pi}{2} -\frac{4}{\pi} \sum_{k\geq 1, k\ odd}^{\infty} ...
2
votes
2answers
521 views

The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series

So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this. I know 'big ell 2' and 'little el 2' are ...
1
vote
1answer
27 views

Writing a Fourier series of a $2\pi$-periodic function.

This problem was taken from Stein's Introduction to Fourier analysis, and it goes like this: Let $f$ be a $2\pi$-periodic Riemman integrable function defined on $\mathbb{R}$. Show that the Fourier ...
1
vote
1answer
42 views

Sum involving the “distance to the nearest integer function”

I want to prove that if $||x||$ is the distance between $x$ and the nearest integer to $x$, $\{\alpha_1,\ldots, \alpha_N\}$ are points in $\mathbb{R}$/$\mathbb{Z}$ and we define $$S(y) = ...
1
vote
2answers
53 views

Integrating a Fourier series

I am trying to integrate the Fourier series of $$f(x) = x,-\pi<x<\pi.$$ Using complex exponentials to find the series, I get the series $$\frac{2}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} ...
1
vote
1answer
40 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
76 views

A question about theorem 2 in de Bruijn's 1950 paper “The roots of trigonometric integrals”

Theorem 2 of de Bruijn's paper titled "The roots of trigonometric integrals" (Duke Math. J., 17 (1950)) is given by: What does it mean by "the function $q(x)$ be regular in the sector...? Does it ...
1
vote
1answer
1k views

Proof of Parseval's Theorem for Fourier Series

Ok so I want to prove the above expression, I substituted the complex fourier series for f and using the fact f may be complex-valued, carried on by representing $|f(x)|^2$ as $f(x)f(x)^\ast$ where ...
1
vote
1answer
331 views

Fourier Series of $f$ on the given interval

my goal is to find the Fourier series of f on the given interval: $$f(x) = \begin{cases} 0, & \text{if } -\pi < x < 0 \\ \sin(x), & \text{if } 0 \le x < \pi \end{cases}$$ I know ...
1
vote
1answer
130 views

Uniform convergence of Fourier Series, how do I check it?

Let $f(x)=x(\pi-x)$, $x\in (0,\pi)$. The function satisfies the Dirichlet conditions so its Fourier series, $S_f$ converges pointwise to $f$. The definition of a Fourier series of $f$ on $[a,a+L]$ ...
1
vote
1answer
246 views

What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
1
vote
0answers
231 views

Fourier Series on a 2-Torus

Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a ...
1
vote
3answers
2k views

Fourier transform of $\frac{\sin{x}}{x}$

can you help me with this question find $\sin$ Fourier transform of $\frac{\sin{x}}{x}$
1
vote
1answer
655 views

Dirichlet kernel.

I have a function $h\in L^1(\mathbb{T})$, and I want to show that: $$\int_{\pi\geq |t|>\delta>0} h(x+t)D_N(t) dt/2\pi \leq \xi_N(h,\delta)$$ where $\xi_N(h,\delta) \rightarrow 0$ as ...
1
vote
2answers
203 views

$f,g$ are two continuous functions with period$=1$, are the Fourier coefficients $f*g=f(n)g(n)$?

Let $f,g$ be two continuous functions with period$=1$. Are the Fourier coefficients of $f*g$ are given by the products $f(n)g(n)$ (of the $n$-th coefficient in each series)? Thanks!
1
vote
1answer
1k views

Convergence of Fourier series for $|\sin{x}|$

I was solving this question I saw in a textbook. The question is : Calculate the Fourier series for $ f(x) = |\sin x| $ for $-\pi \leq x \leq \pi$. Which I had $ f(x) = \frac{a_{0}}{2} + \sum ...
1
vote
1answer
682 views

rapidly decaying sequence and fourier series coefficients of a compactly supported smooth function

In this question the term rapidly decaying sequence is used. What is the definition of a rapidly decaying sequence.(in cases of terms being purely real or complex.) How to prove that the sequence of ...
1
vote
1answer
276 views

Bounds for Fourier series

Fourier series of function f: $$f(x)=\sum_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$$ Suppose $f_{s}\sim\frac{1}{s^{p}}$. What can we say about $f(x)$? Can we find some bounds for $f(x)$ like ...
1
vote
1answer
2k views

Continuity and discontinuity in fourier series?

Can somebody please explain continuity and discontinuity in fourier series?
0
votes
0answers
51 views

Proving Gibbs phenomenon using Dirichlet kernel

I am working on a problem$^{(1)}$ on using Dirichlet kernel to prove Gibbs phenomenon. It is a long proof broken down into 7 steps, and on each step I have to answer some questions. Long story short, ...
0
votes
0answers
29 views

Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
0
votes
1answer
51 views

Computing the Fourier series of $f = \cos{2x}$?

I'm currently attempting to solve the following problem: Given the function $f$ defined on the interval $(0, \pi)$ by $f(x) = \cos{2x}$, find the $2\pi$-periodic, even extension of $f$ and compute ...
0
votes
1answer
78 views

From fourier series to continuous fourier transform

In derivation of fourier transform, we start with the fourier series coefficients. If we let $T \to \infty$, it's common to say the spacing between consecutive fourier coefficient will approach $0$, ...
0
votes
1answer
22 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
0
votes
1answer
90 views

Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
0
votes
1answer
42 views

Want to prove that the Hilbert transform of a $C^1(\mathbb T)$ function is the principal value of the convolution with $\cot(\pi x)$

So here is my problem, Let $L^2_0:=\{f\in L^2: \hat{f}(0)=0\}$ and consider the Hilbert transform given by the following map $$H:L^2_0([0,1])\rightarrow L^2_0([0,1])$$ $$f\mapsto (\mathcal ...
0
votes
1answer
46 views

Check answer on given question

I would like to take care on my answer on the following question Fourier transform involving a dirac delta function I have tried to answer this question,of course did not know exact answer,just if ...
0
votes
2answers
86 views

Using fourier analysis in order to solve differential equations.

http://www.enm.bris.ac.uk/admin/courses/EMa2/Lecture%20Notes%2009-10/LSPDE5.pdf The above PDF teaches us the separation of variables method. However, there are some things I dont understand, that I ...
0
votes
2answers
499 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
1answer
93 views

Fourier expression for infinite recurring binary sequence

We have infinite binary sequences of type $$\langle g_n \rangle_{j=4}=\{0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,...\} \,;\, n\to\infty$$ where $j$ indicates the length of a period. we try to express them ...
0
votes
1answer
221 views

Show $\left\lvert\sum_{k=-n}^n \frac{\sin k t }{k}\right\rvert \le \pi + 2$ for all $n$ and $t$, $t \in [0,2\pi]$

Let $f(t)=(t-\pi)\chi_{(0,2\pi)}$, $t \in [0,2\pi]$, then the partial sum of the Fourier series of $f$ is $$ S_n(t)=- \sum_{0 < |k| \le n} \frac{\sin k t}{k}. $$ Show $|S_n(t)| \le \pi+2$ for all ...
0
votes
4answers
334 views

Let $a_{n}\downarrow 0$ and if series $\sum a_{n}\sin nx$ is a Fourier series of function $f\in L^{1}$ then $\sum \frac{a_{n}}{n}<+\infty.$

I want to show that if $\displaystyle\sum a_{n}\sin nx (a_{n}\downarrow 0)$ is a Fourier series of $f\in L^{1}$ then $\displaystyle \sum \frac{a_{n}}{n}<+\infty.$ I know i have to use some property ...
0
votes
1answer
1k views

Deriving fourier series using complex numbers - introduction

So this is the follow up thread to the one I asked before but you don't need to read the other one for this to make sense. If you want to, read PZZ's answer: link to the thread. So I know that there ...
0
votes
2answers
309 views

Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc?

Consider any continuous $2\pi$ periodic function (of bounded variation) $f : \mathbb{R} \to \mathbb{R}$ and its fourier series given as $f(\theta) = \frac{a_o}{2} + \sum\limits_{n = 1}^{\infty} ...
0
votes
1answer
468 views

Proving that the fourier coefficients for a pretty smooth function are pretty small

Let $f:[0,2\pi] \rightarrow \mathbb{R}$ be $C^k$ for some $k >0$. Prove that $|\widehat{f}(n)|n|^k|$ is bounded above by some constant independent of $n$. To do this, we've been ...