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.

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5
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1answer
123 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 ...
4
votes
1answer
54 views

Closed form for $\int^{\pi}_0 \frac{\sin^2 (y)}{a+\cos(y)} \cos(ny) dy$ for integer $n$

I encountered this integral when trying to obtain a Fourier series for the function inside (in connection to this question). Mathematica gives the following general solution (only valid for ...
4
votes
2answers
161 views

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all ...
4
votes
4answers
468 views

Evaluate $ \sum_{n=1}^{\infty} \frac{\sin \ n}{ n } $ using the fourier series

I am a beginner with Fourier series and I have to evaluate the sum $$\sum_{n =1}^{\infty}{\sin\left(n\right) \over n}$$ I don't know which function I have to take to evaluate the fourier series ...
4
votes
1answer
232 views

Heuristic\iterated construction of the Weierstrass nowhere differentiable function.

I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example ...
4
votes
2answers
332 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
2answers
455 views

“Every function can be represented as a Fourier series”?

It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series. So this got me thinking about the ...
3
votes
0answers
78 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 ...
3
votes
1answer
256 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
3
votes
2answers
59 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
2
votes
2answers
101 views

Why does specifying an interval for a function make the function odd or even?

I am currently reading about Fourier series and Orthogonality of functions and Complete Sets of functions. Below are two extracts from the book I'm reading for which I simply do not understand: ...
2
votes
1answer
94 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
34 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
287 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
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
403 views

Calculating the Fourier series of $x^{3}$

I was given as homework to calculate the Fourier series of $x^{3}$. I know, in general, how to obtain the coefficients of the series using integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
2
votes
1answer
1k views

A function and its Fourier transform cannot both be compactly supported

I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know $f$ is a trigonometric polynomial, I see how to finish the ...
1
vote
0answers
49 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
113 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
78 views

Using Hölder condition to find upper bound on Fourier coefficients

First I want to stress that I don't want an answer, perhaps a hint. Let $f(x)$ have period $2\pi$ and let $|f(x) -f(y)| \leq c|x-y|^{\alpha}$, for some constants $c$ and $\alpha$ for all $x$ and $y$. ...
1
vote
1answer
110 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
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1answer
69 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
64 views

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

With the following Mathematica program: ...
1
vote
1answer
243 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
263 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
0answers
278 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
4k 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
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
0answers
30 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
78 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 ...
9
votes
8answers
1k 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 ...
8
votes
2answers
536 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
6
votes
3answers
877 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
14k 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
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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 ...
5
votes
1answer
5k 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 ...
4
votes
1answer
565 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 $\sum\limits_{n=1}^\infty a_n\sin nu=\cos u$ ...
4
votes
2answers
3k 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
108 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
132 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
64 views

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of ...
3
votes
1answer
675 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
1answer
212 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
72 views

Find the Fourier series of the trigonometric polynomial $f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx})$

I'm learning about Fourier series and need help with this problem: Given the trigonometric polynomial $$ f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx}) $$ find the Fourier ...
2
votes
1answer
83 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
74 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 ...
2
votes
1answer
112 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
183 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
177 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!