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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Fourier Transform, Laplace Transform, but what about…

I have a question regarding the fourier and laplace transform. First, the Fourier transform essentially takes a function, divides it by a frequency (imaginary exponential), and then sees how much of ...
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34 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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1answer
149 views

Fourier series of f(x)

I want to find the Fourier series of $f(x)$ defined by $f(x)=\begin{cases} 1 , -L\le x<0\\ 0, 0\le x<L. \end{cases} $ Well, to find $a_0$ I do this integral: $$a_0=1/L \int _{-L}^0 dx +1/L ...
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1answer
35 views

Discrete Fourier Series

I have a series of discrete values that are periodic and I am looking to calculate the Fourier series of it. I learnt all of this in college but I can't for the life of me remember now. The discrete ...
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1answer
18 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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28 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
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61 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
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3answers
31 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
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2answers
58 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
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1answer
27 views

How to compute Fourier coefficients using a cubic spline-corrected FFT?

I'm not particularly experienced in numerical analysis, and so I recently had quite a massive shock when I discovered that sampling a smooth function and computing the FFT of the result does not ...
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1answer
55 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
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1answer
36 views

When do the sine components of a Fourier series vanish?

A Fourier series is given by: $$ s_N(x) = \sum c_n \cdot e^{i \frac{2\pi n x}{P}} $$ With Euler's identity, the exponential can be converted to a sums of sines and cosines. When do the sine ...
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3answers
193 views

Why is $\sum_{n=-\infty}^{\infty}\exp(-(x+n)^2)$ “almost” constant?

I did some numerical approximation of $$\sum_{n=-\infty}^\infty \exp(-(x+n)^2)$$ and found that this function is "almost" constant ($\approx 1.772$). Why does the sum fluctuate little? Is there a ...
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1answer
33 views

What is the Fourier series of $\frac1T\sum^{\infty}_{m=-\infty}\delta(f-\frac mT)$?

As the title mentioned, I've not known exactly about Fourier series and when I was reading an digital communication textbook, I wondered about below equation derivation of Fourier series like ...
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4answers
77 views

a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
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1answer
34 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
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1answer
35 views

A problem concerning finite number of Fourier coefficients

Is there a smooth, non-zero $2\pi$-periodic function $f,$ with support of $f$ contained in an interval $[a,b]\subset[0,2\pi],$ such that $b-a<2\pi$ and only finitely many Fourier coefficients of ...
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1answer
30 views

Prove the uniform convergence of a Fourier series

Suppose that $f$ is a $2\pi$-periodic function that satisfies the estimate $$|f(x)-f(y)|\leq M|x-y|^\alpha$$ for an $0<\alpha<1,$ and let ...
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13 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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68 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
27 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
87 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
16 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
51 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
22 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
39 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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39 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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81 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
37 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
62 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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1answer
31 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
23 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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10 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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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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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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1answer
36 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
45 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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1answer
79 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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1answer
52 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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39 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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1answer
26 views

Convolution theorem for product of functions

I like to Fourier transform the following product of functions: $$g(\vec{r})f(\vec{r}).$$ So I like to calculate the following: $$\int g(\vec{r})f(\vec{r}) e^{-i\vec{k}\cdot\vec{r}}d^3r.$$ ...
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7 views

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
72 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
49 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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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
49 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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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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34 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 ...