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
1answer
64 views

Why do the first spikes in these plots point in opposite directions?

With the following Mathematica program: ...
1
vote
1answer
418 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
1
vote
1answer
2k 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 ...
0
votes
1answer
60 views

Partial Sum Fourier Series

Show that the partial sum $$f_N(x)=\frac{4}{π}\sum^N_{n=1}\frac{\sin((2n-1)x)}{2n-1}$$ may be written as $$f_N(x)=\frac{2}{π}\int_0^x\frac{\sin(2Nt)}{\sin(t)}\,dt$$ The original question is 'Sketch ...
6
votes
1answer
323 views

Cusp forms' Fourier coefficients sign changes

I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N ...
5
votes
1answer
112 views

Fourier Series: going from $a_n$ and $b_n$ to $c_n$

I sort of understand the principle of the Fourier series, but when I watch the wiki page I don't understand how to get from: ${a_0 \over 2} + \sum_{n=1}^N[a_n cos({2\pi n x \over P}) + b_n sin({2\pi ...
5
votes
1answer
4k views

Fourier series for $\sin x$ is zero?

I have no practical reason for wanting to do this, but I was wondering why the Fourier series for $\sin x$ is the identical zero function. I'm probably doing something wrong or missing some important ...
3
votes
2answers
518 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
2answers
293 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
206 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 ...
2
votes
1answer
91 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
2
votes
2answers
32 views

Removing $e^{-in\pi x/\ell}$ from an integral

I'm considering a proof of the convergence of the Fourier series. It begins by considering the full Fourier series of the periodic extension of $\phi$ defined on $[-\ell, \ell]$. The full Fourier ...
2
votes
1answer
251 views

Derivation of fourier series equation

No matter where I search, every time if there's an article about Fourier series derivation, the first step made by author is to present the following formula: $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty ...
2
votes
0answers
69 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
2
votes
1answer
109 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
1k 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 ...
2
votes
3answers
211 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
2answers
616 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
0answers
45 views

Expanding Fourier Series of $f(x)=x^2$ where $0<x<1$ (even and odd)

I tried to solve Fourier series (which appeared on title) and ended up to below solution : on even state : $ \phi(x)= \begin{cases} x^2 & 0<x<1 \\ x^2 & -1<x<0 \end{cases} $ ...
1
vote
1answer
110 views

Fourier series for square-wave function?

Show that the Fourier series for the square wave function $$f(t)=\begin{cases}-1 & -\frac{T}{2}\leq t \lt 0, \\ +1 & \ \ \ \ 0 \leq t \lt \frac{T}{2}\end{cases}$$ is ...
1
vote
1answer
97 views

“Counterexample” for a weaker version of Riemann–Lebesgue lemma

My teacher gave us this version of Riemann–Lebesgue lemma in class: Let $g(t)$ be an absolutely integrable function on $[a,b]$, then $$\lim_{p\to\infty} \int_a^b g(t)\sin(pt)dt=0$$ Similarly for ...
1
vote
1answer
57 views

Fourier series and transform (epicycles)

Let $\gamma:[a,b]\to\mathbb{C}$ be a continuous curve. 1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and some function $\varphi:\mathbb{R}\times ...
1
vote
1answer
32 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
0answers
82 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
209 views

Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
1
vote
1answer
259 views

Green' s function for harmonic oscillator

Does someone know how to get a solution of differential equation for Green's function $(-d^2/dt^2 + \omega^2) G(t, s) = \delta(t-s) $? There is a periodicity of G, actually $\Delta (t-s) = G(t,s)$ ...
1
vote
1answer
751 views

The Fourier series of a continuously differentiable function converges absolutely

Suppose $f$ is continuously differentiable on the unit circle. Show that the Fourier series of $f$ converges absolutely (thus uniformly) to $f$. Let the Fourier series of $f$ be given by $$ f(x) ...
1
vote
1answer
149 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
130 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
2answers
16k 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
820 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 ...
0
votes
0answers
25 views

Show that the series converges and is equal to the following.

Define $F(x) = i(-\pi - x)$ if $-\pi \leq x < 0$ and $F(x) = i(\pi - x)$ if $0<x\leq \pi$, with $F(0) = 0$. Show that if $x \neq 0$ mod $2\pi$, then the series $E(x) = \sum_{n=1}^{\infty} ...
0
votes
2answers
69 views

Verify that $\frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+…$ can be found via a Fourier series for $x$ in $-\pi\lt x \le \pi$

Using the general Fourier series expansion: $$f(x)=\frac{a_0}{2}+ \sum_{r=1}^{r=\infty}\left(a_r\cos\left(\frac{2\pi r x}{T}\right)+b_r\sin\left(\frac{2\pi r ...
0
votes
1answer
49 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
1answer
103 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
248 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
343 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
358 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} ...
9
votes
8answers
942 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
3answers
709 views

FFT with powers of 3

Classic Fast Fourier Transfrom (FFT) works fine, when $n$ is power of 2. How to generalize FFT procedure when $n$ is power of 3? Is it possible to easily modify the algorithm and preserve its ...
6
votes
1answer
12k 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
6
votes
2answers
2k 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 ...
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
96 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
119 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
1k 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
1answer
566 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 ...
2
votes
1answer
66 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
0answers
51 views

Drawing a continuous function with divergent Fourier series at $x=0$…

Does anyone know how the graph looks like for a continuous function with Fourier series diverging at $x=0$ ? The example due to Fejer (a variation of the du Bois-Reymond construction), is explicitly ...