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.

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1answer
41 views

Pointwise convergence of sine series of $x^{-2}$

I was wondering if the sine series of $x^{-2}$ converges pointwise on the open interval $(0,1)$. What is the most general criterion to ensure pointwise convergence of a Fourier series?
2
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1answer
74 views

Computation of $\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy$

Find the value $$\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy,\qquad k\in N^{+}$$ My idea: \begin{align} ...
1
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1answer
61 views

Fourier Series Convergence

Going over some revision. Not really sure what to do for the last bit of aii) I know at $x = 0$, it will converge to $0$ and at $x = \frac{M}{2}$ it will converge to $1$, I'm not seeing how this ...
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1answer
59 views

Calculating value of the fourier series

In the following fourier series, how the red marked numbers are calculated?
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1answer
77 views

How to calculate integral of fourier series?

I know how to integrate, but I can't understand how the integral of this fourier serie is calculated. my problem is with integral of the sigma. fourier: integral: Can anyone say me how this ...
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0answers
311 views

How can I find the compact trigonometric Fourier series from these signals?

I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
3
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1answer
59 views

How to calculate $\sum_{k=1}^{\infty}\frac{1}{(a^2+k^2)^2}$ after calculating $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Parseval identity?

The task is to calculate sum $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Fourier coefficients of $f(x)=e^{ax}$. First of all I calculated Fourier coefficients of the sum: ...
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5answers
447 views

Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
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1answer
201 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
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2answers
778 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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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 ...
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1answer
82 views

Convergence of Fourier series - strange graph in proof

I am reading a text that states the following related to convergence of Fourier series: $$g_K(x) = > ...
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3answers
5k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
2
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3answers
289 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 ...
3
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1answer
79 views

Limit of Multivariable Fourier Series

If I have some Fourier Series representation of a function with $x$ period of $2L$ $$G(x,y) = \sum_{n = 1}^{\infty} \left[a_n \sin\left(\frac{n \pi x}{L}\right) + b_n \cos\left(\frac{n \pi ...
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0answers
230 views

Double Fourier Series $\cos(nx)\cos(my)$

Let $f(x,y) = xy$ on the square $[0, \pi]^2$. Find the Fourier cosine-cosine series of $f$. I am working on this question with a group and one of us gets all the coefficients as zero. Is this correct ...
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2answers
130 views

Find the Fourier series of $g (x) = f (x-a)$, where $f$ is $2\pi$-periodic and $a$ is a real number.

Find the Fourier series of $g (x) = f (x-a)$, where $f$ is $2\pi$-periodic and $a$ is a real number. This is for real analysis so I cannot use Euler's formula to compute the Fourier coefficients.
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1answer
613 views

Fourier Series of Multivariable Functions.

If I have some function $V(x,y)$ which is periodic in x with period L. I wish to expand $V(x,y)$ in terms of a fourier sine (for simplicity) series in $x$, is it always the case that I may write the ...
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0answers
234 views

Taylor series of Fourier series of triangle wave

Odd triangle wave $\text{t}(x)$ with angles at $(2x+1)\in\mathbb{Z}$ can be represented by Fourier series: ...
0
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1answer
99 views

Fouries series of $\sin{\sum_n a_n \sin{n\theta}}$

What's the Fourier series for $\sin({\sum_n a_n \sin{n\theta}})$?
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0answers
318 views

Parseval's formula application [closed]

I need to show that $$1+\frac{1}{3^2}+\frac{1}{5^2}+\cdots=\frac{\pi^2}{8}$$ by evaluating $\|Sq\|^2$ such that $Sq(t)=\frac{4}{\pi} \sum \limits_{n \text{ odd} \geq 1} \dfrac{\sin nt}{n}$ We know ...
4
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1answer
123 views

Fourier series of a function

Consider $$ f(t)= \begin{cases} 1 \mbox{ ; } 0<t<1\\ 2-t \mbox{ ; } 1<t<2 \end{cases}$$ Let $f_1(t)$ be the Fourier sine series and $f_2(t)$ be the Fourier cosine series of $f$, ...
2
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1answer
570 views

Laplace heat equation

An infinite straight metal pipe has annular cross-section $a \leq r \leq b$. The temperature of the inner surface of the pipe is equal to $\cos(\theta)$, and the outer surface is thermally insulted. ...
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2answers
222 views

Fourier Series of a function: best approximation in the sup norm as well?

Prove or disprove the following: Let $V$ be the space of continuous functions on $[0,2\pi]$ with $f(0)=f(2\pi)$. Let $\|f\|=\text{sup}\{|f(x)|: 0\leq x \leq 2\pi\}$. Let $N$ be a positive integer and ...
2
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1answer
75 views

Fourier coeficients, convergence of the integral

Let $g\in L_{\infty}(\mathbb{T})$. For any $-\pi<a<b<\pi$ let $\chi_{[a,b]}$ be the characristic function of $[a,b]$. Prove that ...
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2answers
227 views

Prove that if $\sum c_n e^{inx}$ converges in $L^2$ to $f$ then $c_n$ are the Fourier coefficients.

The full problem is this: Let $f:[-\pi,\pi]\rightarrow\mathbb{C}$ be Riemann integrable. Let $\{c_n\}_{n\in\mathbb{Z}}\subset\mathbb{C}$. Prove that if $s_N=\sum_{n=-N}^Nc_ne^{inx}$ converges in ...
3
votes
2answers
140 views

Identity involving partial sums of Fourier series

Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$ How can I show that ...
0
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1answer
399 views

Fourier series odd extension (with only odd terms)

I'm currently working on this question: Suppose a function is defined only on $[0,L]$. Show that we can write $f(x) = \displaystyle\sum_{n=0}^{\infty} c_n \sin \left(\frac{(2n+1)\pi x}{2L} ...
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2answers
152 views

Convergence of $\sum\limits^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$

Ok, for the infinite series: $$\sum^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$$ How do I show that this converges on any finite interval if $\sum^\infty _{k=0} k(|a_k|+|b_k|)<\infty$? Also, do the ...
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0answers
50 views

Discrete time fourier transform of partial sum

I came across the following property of the DTFT: $ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
1
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2answers
569 views

Proving that two periodic functions are orthogonal

Suppose we have a periodic function $f_K(\vec x)$. We want to show that $\int {f_K^*(\vec x) f_{K'}(\vec x) d\vec x} = \delta_{KK'}$, where the integration is over the period of $f(x)$. I know this ...
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1answer
92 views

Alternative complete bases for Fourier Series.

Knowing that $$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$ are complete systems in $L^2(0,\pi)$. How ...
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2answers
60 views

Integral of a special Fourier Serie into the hipercube of s dimension.

Given the numbers, $u\in\mathbb{R}^s$, $\alpha>1$ and $s>1$. If we have the below Fourier Serie: $f_{\alpha}(u)=\sum\limits_{h\in\mathbb{Z}^s}\frac{1}{r(h)^{\alpha}}\exp^{2 \pi i ...
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0answers
135 views

Prove a function has $k$ continuous derivatives from its Fourier series

Here is the problem. Let $k\in \mathbb{N}$. Suppose that there is a constant $C$ such that $|c_n|<\frac{C}{|n|^{k+1}}$ ($c_n$ here is the $n$th Fourier coefficient). Prove that ...
3
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0answers
459 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
5
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3answers
250 views

A question related to Wave Equation

Let $L>0$. Suppose $f, g$ are $C^2$ functions on $\mathbb{R}$ such that $$f(t)+f(-t)+\int_{-t}^t g(s)\,ds=0$$ and $$f(L+t)+f(L-t)+\int_{L-t}^{L+t} g(s)\,ds=0$$ for all $t\in \mathbb{R}.$ Does it ...
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0answers
135 views

Eigenfunction of the $n$-torus

Let $\phi$ be an eigenfunction of the Laplacian $\Delta$ on the $n$-torus $T^n$, with eigenvalue $-\lambda$, i.e. $\Delta \phi + \lambda \phi =0$, then : $$ \phi (x)= \sum_{|n^2|=\lambda} \hat{\phi} ...
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0answers
233 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
55 views

Fourier series inequality with polynomial

I have the following question: Let f be in $\mathbf{L}_{\mathbf{R}}^2([-\pi;\pi])$. Show that $$\left({\int_{-\pi}^\pi |x^nf(x)|\,\mathrm{d}\lambda(x)} \right ) \leq \frac{2*\pi^{2n+1}}{2n+1} ...
3
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1answer
49 views

Analysis: Show that $f(x)$ converges, pointwise and uniformly on $\mathbb{R}$ to a differentiable function $f$ that satisfies:

Show that $$f(x) = \sum_{k=1}^\infty \frac{\sin kx}{k^3} \tag1$$ converges, pointwise and uniformly on $\mathbb R$, to a differentiable function that satisfies $$\int_0^{\pi/2} f(x)\,dx = ...
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4answers
402 views

Type of convergence of Fourier series of a function

For given function, $$f(x) = \begin{cases} 1, & |x|\leq\frac{\pi}{2} \\ 0, & \pi\geq|x|>\frac{\pi}{2} \\ \end{cases}$$ The calculated Fourier series is: $$\begin{align} a_0 &= ...
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1answer
86 views

Approximation using a Fourier transform with low pass filter

I need to approximate a function f, but I cannot do so with frequencies that exceed 1kHz What is the best approximation I can get? Is taking the Fourier transform then zeroing any term above 1kHz the ...
5
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0answers
179 views

Expansion in Fourier series involving a complicated “argument”

I know how to expand a function $f(x)$ into a Fourier series with the period $2L$: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos(n\pi x/L)+\sum_{n=0}^\infty b_n\sin(n\pi x/L),$$ but what if I ...
2
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0answers
61 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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2answers
200 views

Does $\sum_{n=0}^\infty\frac{\sin(2n+1)}{2n+1}=0$?

I've come to a bit of a sticking point in my answer to problem 14A given here http://www.maths.cam.ac.uk/undergrad/pastpapers/2011/ib/List_IB.pdf (note that this is a past paper that I am trying for ...
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1answer
83 views

$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0 }\sum_{n=-\infty}^{\infty}\delta f(\delta n)$?

Assume that $f$ is continuous and moderate decrease, show that $$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0,~ \delta>0}\sum_{n=-\infty}^{\infty}\delta f(\delta n)$$ From the ...
5
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0answers
317 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
3
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1answer
802 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, ...
4
votes
1answer
290 views

Fourier Series and Inner Product

When working with Fourier series, the inner product is defined as $$\int_{-L}^L f(x)g(x)dx$$ I see this definition everywhere and we know that $\rm{sin}\big(\frac{n\pi x}{L}\big)$ and ...
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2answers
383 views

Trigonometric Identities and Fourier Series

I have the series: $$2+\sum_{m=1}^n 4(-1)^m\cos(m\pi x)$$ Here, $x\in (-1,1)$. I need to show that this equals some fraction with only cosine terms and $n$ (no $m$). Just looking for some ...