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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1answer
235 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
1answer
697 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
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 ...
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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 $k\in\...
3
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0answers
82 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
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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 \ell^2(\...
3
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3answers
73 views

Deriving formula from Fourier series: $\frac{\pi^2}{12} = \sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$

The equation/formula $$ \frac{\pi^2}{12} = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$$ is to be derived. I know that the Fourier expansion of $f(x)=x$ for $x \in (-\pi,\pi)$ is $$f(x)=x=\sum_{n=1}^{\...
3
votes
1answer
268 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, ...
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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 ...
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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
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1answer
42 views

Show that $\lim_{n\to\infty}\|f_*^{(n)}\|_1^{\frac{1}{n}}=\|\hat{f}\|_\infty$

Let $1<p\leq 2$ and $f\in L^p(\mathbb{T})$, i.e. $f$ is $p-$th power integrable and is $1-$periodic. Define $$f_*^{(n)}=f*f*\dots*f\quad n\text{ times}$$ Show that $$\lim_{n\to\infty}\|f_*^{(n)}\|...
2
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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
3answers
406 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 ...
1
vote
1answer
115 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 $$f(t)=\frac{4}{\pi}\...
1
vote
1answer
247 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 $F^{-1}DF$ ...
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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} $ $...
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0answers
284 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 ...
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1answer
115 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 $\...
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1answer
64 views

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

With the following Mathematica program: ...
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3answers
4k views

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

can you help me with this question find $\sin$ Fourier transform of $\frac{\sin{x}}{x}$
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1answer
2k 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 & \...
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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} \frac{...
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2answers
79 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 x}{T}\right)\right)\space\space\space\...
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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
551 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 = \dfrac{\...
6
votes
1answer
15k 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
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3answers
899 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 ...
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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 "...
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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 ...
5
votes
1answer
251 views

For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero

Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. I'...
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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 ...
4
votes
1answer
581 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
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1answer
217 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 ...
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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
684 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
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
139 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, while ...
2
votes
1answer
86 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 ...
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0answers
84 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 ...
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3answers
234 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
681 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 ...
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3answers
650 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
2answers
186 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 $]-\pi,\pi[$ ...
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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!
2
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1answer
113 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 $$\...
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votes
1answer
7k 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} \frac{cos(kx)...
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vote
2answers
219 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!
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1answer
3k views

Continuity and discontinuity in fourier series?

Can somebody please explain continuity and discontinuity in fourier series?
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2answers
922 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 ...