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
2answers
929 views

Fourier coefficients of the product of two functions

Given two functions $f,g\in L^2(\mathbb{T})$, I have to prove that the Fourier coefficients of $fg$ are given by $$\hat{fg}(n)=\sum_{k\in{Z}}\hat{f}(n-k)\hat{g}(k)$$ and that this series converges ...
1
vote
2answers
12k views

How to find inverse Fourier transform

I have the function $$ \delta(f-2) $$ How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$. The correct answer is $$ e^{4\pi i ...
1
vote
1answer
1k views

Fourier transform from limit of fourier series [duplicate]

Possible Duplicate: Derivation of Fourier Transform? How is the Fourier transform obtained by taking the limit of the Fourier series as the period goes to infinity? In particular I am ...
9
votes
7answers
770 views

Why does this Fourier series have a finite number of terms?

I am learning about Fourier series in class and the basic form of a Fourier Series is $$a_{0}+\sum_{n=1}^{\infty} [a_{n}\cos(nx)+b_{n}\sin(nx)]$$ so a fourier series should have an infinity number ...
6
votes
2answers
1k views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
5
votes
1answer
9k views

What is the Fourier transform for $f(x)=e^{-x^2}$

I remember their being a special rule for this kind of function but I cant remember what it was. Anyone know how ? thanks
4
votes
2answers
2k views

Identifying the product of two Fourier series with a third?

Given the product of two functions defined explicitly through their Fourier coefficients (of unknown undeveloped form): $\sum_k{c_k e^{i k t}} \cdot \sum_k{c'_k e^{i k t}}$ Is there any way to ...
3
votes
3answers
87 views

Fourier series simplification

I want to show that $$\frac{1}{\pi} \int_{-\pi}^{\pi} f(x)g(x)dx = \frac{a_0\alpha_0}{2} + \sum_{n=1}^{\infty} (a_n\alpha_n + b_n\beta_n)$$ where $f,g: [-\pi,\pi] \to \mathbb{R}$ are integral ...
3
votes
1answer
118 views

Fourier series for $[x]-x+\frac{1}{2}$

$[x]-x+\frac{1}{2}$ has the Fourier series $$\sum_{n=1}^{\infty} \frac{\sin{2n\pi x}}{n\pi}.$$ By evaluating the series directly, which requires some work, it can be shown that the series is ...
3
votes
1answer
863 views

Fourier and Legendre series

Find the Fourier sin series for the function $f(x) = x^3$ on the interval $0\leq x \leq L$. the Legendre series for the same function. One representation involves an infinite number of terms, ...
3
votes
2answers
447 views

Is it possible for cosine functions to have Fourier sine series expressions or sine functions to have Fourier cosine series expressions?

Is it possible for cosine functions to have Fourier sine series expressions or sine functions to have Fourier cosine series expressions? For example, do $\sum\limits_{n=1}^\infty a_n\sin nu=\cos u$ ...
3
votes
1answer
427 views

heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
3
votes
2answers
277 views

What is the reason for these jiggles when truncating infinite series?

Plotting the series $$\displaystyle y = \sum_{k} \frac{\sin kx }{k}$$ In the limit it would look like Taking a finite number of terms, I want to understand what is the reason for the jiggling at ...
3
votes
1answer
192 views

How large are the second, third, fourth, etc. ringing artifacts in Gibbs phenomenon?

I've read that in the Gibbs phenomenon, partial Fourier series will over- or underestimate a function's value in neighborhoods of jump discontinuities. Specifically, the maximum error will converge to ...
3
votes
2answers
3k views

Fourier cosine transform

Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$. I can solve this question only if there is $x$ instead of $x^2$ in the exponential ...
2
votes
1answer
48 views

Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for ...
2
votes
1answer
101 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
2
votes
2answers
160 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
132 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
194 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
492 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
565 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
66 views

Find the half range cosine fourier series expansion for $f(x)=(x-1)^2,\quad 0<x<1$.

Find the half range cosine fourier series expansion for $$f(x)=(x-1)^2,\quad 0<x<1$$ and hence deduce that $$\pi^2=8\left(\frac 1 {1^2}+\frac 1 {3^2}+\frac 1 {5^2}+\ldots\right)\tag{1}$$ My ...
1
vote
1answer
34 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
100 views

Fourier sine series expansion

The function $f(x)$ is defined as $$f(x)=1\qquad0<x<\pi$$ Sketch the odd extension and show that the Fourier sine series expansion is ...
1
vote
1answer
65 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
58 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
26 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 ...
1
vote
1answer
48 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
2answers
39 views

What is the $L$ in the Fourier series term?

I am a bit confused about this: I want to calculate the Fourier series $S^f$ of $f(x)$, where $f$ is periodic with period $k\in \mathbb{R}$. I know that the equations for my terms are: ...
1
vote
0answers
78 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
405 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
137 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
103 views

Fourier Series for $|x|$

How can I calculate the Fourier series for |x| (where $x\in[−\pi,\pi]$) in the complex form? Thanks.
1
vote
1answer
272 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
240 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
3k 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
128 views

Question about Fourier series

The Fourier series of a function $f: G \to \mathbb C$ where $G$ is a group is the representation of $f$ in terms of characters $\chi_g \in \mathrm{Hom}(G, S^1)$ of $G$. I understand the case where ...
1
vote
1answer
719 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
213 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
2answers
786 views

Fourier transform of $\mathrm{sinc}(4t)$

I'm preparing for an exam in the signals and systems class I'm taking. One of the practice exams has a problem that requires you to take the Fourier transform of $\text{sinc}(4t)$. From a table of ...
1
vote
1answer
732 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
285 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
62 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
43 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
57 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 ...