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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Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
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4answers
187 views

Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

I am trying to solve exercises for the coming exam, and I am stuck on this exercise: Evaluate $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
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8 views

Fourier series solution

Show that Summation (1/n^2) = pi^2 /6. Using fourier series. This is the question in BS grewal for fourier series. I dont understand how to apply fourier series without a limit. (-pi to pi ) or to 2pi ...
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1answer
56 views

Show $\lim_{n\to\infty} n^p f(nx) = 0$ exists in the distributional sense

Let $f\in C^\infty(\mathbb R)$ be periodic, with period $2\pi$ and have mean zero ($\int^{2\pi}_0 f(x)dx =0$). Show that for any positive integer $p$ the following limit is valid in the ...
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1answer
45 views

Complex Fourier Series and using the square norm

Find the complex Fourier series of $f(x)=e^{(-πx/2)}$ on $-π < x < π$ Discuss the significance of $|C_n|$ in the solution. I've tried so far Using the Complex Fourier Series: $$ %% ...
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1answer
21 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
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1answer
84 views

Fourier Series $\sin(\sin(x))$

Can anyone find the Fourier Series of $ \sin(\sin(x))$? I have tried evaluating the integrals to determine the coefficients of each of the coefficients of the sine waves, but have no idea where to ...
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1answer
15 views

How to visualize projection of a function onto fourier basis?

I wonder if there are any notes on how one would visualize a projection of function f(x) onto cos(x) and sin(x) in the same way that you would for two vectors. Is there a picture, or a figure ...
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1answer
20 views

The bandwidth of the signal $x(t)$.

The bandwidth (B) of the signal $x(t)$ is the range of frequencies (measured on the positive semi-axis) in which $X(\omega)$ takes values ​​different from $0$. Very often $X(\omega)$ is different from ...
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1answer
26 views

Show solution to ODE's fourier series is a series of sines only

This question was given in an exam in applied mathematics, on the subject of Fourier series: Observe the following ODE: $u\left ( x \right) ^{\prime \prime}+Q \left ( x \right) u\left ( x \right) ...
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1answer
20 views

Shortcut to sine series using regular expansion?

If we're given the Fourier series of $e^x$ on the interval $(0,2\pi)$, I'm wondering if there's a nicer way to extract the sine series of $e^x$ on the same interval other than getting the coefficients ...
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1answer
37 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
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1answer
75 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
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1answer
22 views

Not understanding one step in derivation of Dirichlet kernel

I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the ...
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0answers
36 views

Real-valued Fourier series representation

I have got stuck on the following task: Find the value of the series $${4\over \pi^2}\sum_{k=1}^\infty {1\over k^2}-{1\over \pi^2}\sum_{k=1}^\infty{(-1)^k\over k^2}$$ using real-valued Fourier ...
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78 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
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1answer
28 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
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1answer
33 views

Interpreting Fourier transform frequency graph

I've been trying to understand Fourier transform for some time now and I think I've perhaps finally got the idea now. What I would like to do now is to make an example of Fourier transform for ...
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0answers
12 views

Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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2answers
59 views

what is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$

Can anyone help me with this? What is the sum of this series: $\frac{2}{\pi}\Sigma_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}$ I got it after plugging $x=-1$ in a fourier series Thank you!
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4answers
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Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
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0answers
9 views

Fourier expansion of sum of an arbitrary function and a trig function

I have this BVP with initial condition being $v(x,0) = -x/\pi - (1/25)sin5x$ and I'm looking for $v(x,t) = \sum b_n sin(nx)e^{-n^2t}$ Expanding $v(x,0)$ gives $v(x,0) = -(1/25)sin5x - ...
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1answer
41 views

How to obtain a periodic function from a rapidly decaying function?

Suppose $f(x) = \exp(-x^2)$ with $x \in [0, 3]$. How could I periodise this function to obtain an analytical form of a continuum periodic function $x \in [0, +\infty)$ with period T = 3?
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1answer
42 views

Fourier series of oscillation in form $\cos(2 \pi \frac{k}{T}+\phi)$

I would like to calculate the fourier coefficients of $\cos(2 \pi \frac{k}{T}+\phi)$ where $T \in \mathbb{N}$ is the period and is arbitrary but fixed, $k \in [1, N-1]$ is the number of oscillations ...
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2answers
31 views

Filter on Fourier Series

i have a lowpass filter H(ω) which is $ H(ω) = e^{-jω} $ on -2π≤ω≤2π, and $0$ elsewhere and i have a function in fourier series y(t), i need to find the new signal (z(t)) after the application of the ...
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0answers
49 views

Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
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1answer
77 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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2answers
120 views

Condition on Fourier coefficients for real-valued function

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is real-valued if and only if ...
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1answer
49 views

Will Fourier Series converge even if you only use Prime Integer frequencies?

So there is a Fourier Series for a function $f(x)$ with period $P$: $$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(\frac{n 2 \pi x}{P} + \phi_n \right) $$ Let $\frac{2 \pi x}{P} = t$ ...
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1answer
34 views

Subtracting terms from a Fourier series

It is known that $\sum_{n=1}^{\infty}\frac{\sin(nx)}{n}=\frac{\pi-x}{2}$ in $]0,\pi]$, mostly because this is a way of evaluating $\zeta(2)$. Knowing this, is there a way to evaluate ...
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1answer
37 views

How can we represent an image using basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But i don't understand terms like "orthogonal " and "basis ...
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Are the dominant frequencies preserved under fractional inversion

Let $f(t)$ be a signal that is a function of time. Let $F(f)=\mathcal{F}\{f(t)\}$ be the Fourier transform of $f(t)$. If $F(f)$ is dominated by a sparse set of frequencies $(f_1,f_2,\cdots,f_n)$ (only ...
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2answers
65 views

Wolframalpha find Fourier series

Is there a way to write down the following in Wolframalpha? $$f(x)=\begin{cases}1-x,& 0\leqslant x\leqslant 1\\ 0,&1\lt x\leqslant2\end{cases}.$$
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1answer
41 views

Proof of Fourier series Theorem (k-continuous derivatives)

Here's the theorem: Theorem: If $f$ is periodic with Fourier coefficients $a_n,b_n$ and if the series $$\sum_{n=1}^\infty (|n^{k}a_n|+|n^{k}b_n|)$$ converges for some integer $k \geq 1$, then f ...
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0answers
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Jacobi Form and its Fourier expansion

Let k,m be non negative integers. A Jacobi form of weight k and index m is a holomorphic function f on $\mathbb{H} x \mathbb{C}$ (where $\mathbb{H}$ denotes the upper half plane) satisfying the ...
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1answer
46 views

Series expansion of $\coth x$ using the Fourier transform

Hi I have research about the series of coth but all of the solutions emerges from integral on a contour, Could you calculate the fourier transform of coth? Is that possible at all?My goal is to reach ...
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31 views

Relation between permutations and fourier transform?

i dont know if this is already addressed somewhere (searching around did not find sth). The motivation is to find a way to generate or produce permutations using concepts from continuous mathematics ...
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1answer
22 views

Evaluate the series $\sum_{n = 0}^\infty \frac{1}{(2n + 1)^6}$ by examining the real Fourier series of the function $f(x) := x(\pi - |x|)$

The following is a question from a past exam in my university in a course called "Mathematical Methods for Statistics". It consists of two subquestions that may or may not be related (there is a high ...
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2answers
50 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
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32 views

Finding Fourier series of $x(a-x)$ by integrating the Fourier series of the delta function.

I want to find the Fourier series for $$f(x)=x(a-x).$$ Of course I could do integration by parts and find the coefficients that way, but I'm given a hint to integrate the Fourier series expansion of ...
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19 views

Is there addition theorem for Fourier Harmonics?

We know that in spherical harmonic expansion we have addition theorem, and we can expand a function which depends on $x,x'$ and the angle between thesis two vectors $\cos(\theta_{x,x'})$ by spherical ...
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1k views

Performing a differentiation on a Fourier series

I'm tutoring a set of problem sheet to do with Fourier series and one problem is as follows: The Fourier series for a sawtooth wave is, $f(x)=x=-\sum^{\infty}_{n=1}\frac{2(-1)^n\sin(nx)}{n}$ ...
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1answer
19 views

N-point FFT and 2-radix FFT

I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length $2^n$) For example a is a sequence: ...
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66 views

How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
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1answer
28 views

$ U_{xx}+U_{yy}=0$ with rectangular boundary conditions

When solving $ U_{xx}+U_{yy}=0$ with $u(0,y)=u(a,y)=u(x,b)=0,u(x,0)=f(x)$. $0<=x<=a$ , $0<=y<=b$ by the method of separation of variables I have $-X''(x)-\lambda X(x)=0 $ ...
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1answer
46 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
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1answer
42 views

Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that ...
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1answer
55 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
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2answers
32 views

Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
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1answer
29 views

Is this fourier even?

$$ f(x) = \begin{cases} \frac{\pi}{4}-\frac{x}{2} & [0,\pi] \\ -\frac{3\pi}{4}+\frac{x}{2}, & (\pi,2\pi) \end{cases} $$ Is it right to compute only $a_n \text{ and } a_0$ coefficient for ...