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Questions tagged [fourier-series]

A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.

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Prove that the alternating sum of the odd reciprocals is $\frac{\pi}{4}$ using Fourier and Parseval's theorem [on hold]

Using the Fourier series of the function $f(x)=-k$ if $-\pi<x<0$ and $f(x)=k$ if $0<x<\pi$ and applying Parseval's theorem, prove that: $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\...
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Difference real and complex fourier series

I'm working on fourier series and I'm trying to compute the fourier transformation for the $2\pi$-periodic function of $f(x)=x^2$ with $x \in [-\pi,\pi]$. Now with the real way, that is $$f(x) \sim \...
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31 views

Simpson vs. trapezoidal rule for numerically integrating $\cos{x}\cosh{x}$ in range 0 to $\pi$?

I have to numerically calculate many integrals similar to this: $$\int_0^\pi \cosh{\left(\frac{a_1\cos{x}+a_2\cos{2x}+a_3\cos{3x}+\ldots}{10}\right)}\cos{jx}\cos{kx}\space dx$$ Right now I am using ...
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1answer
24 views

Find a trig polynomial, $(P_N)_N$, s.t. $(P_N)_N\rightarrow |x|$ on $\Big[-1/2,1/2\Big]$

Let $f$ be the $1$-periodic function such that $f(x) =|x|$ for $x \in \Big[ \frac{-1}{2},\frac{1}{2} \Big].$ Determine explicitly a sequence of trigonometric polynomials $(P_N)_N$ such that $P_N \...
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Elliptic Fourier Descriptors - how to derive the coefficients?

In Kuhl and Giardina's 1982 paper, it derives the coefficients, $a_n$ and $b_n$, by equating the two definitions of $\dot{x}(t)$: $$ \begin{align*} \dot{x}(t) &= \sum_{n=1}^\infty \alpha_n \cos\...
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24 views

When Fourier coefficients are real?

Consider a periodic piecewise continuous complex valued function defined on the real line, under what conditions its Fourier coefficients are all real? Or, can they be complex?
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29 views

Fourier series coefficients for a piecewise periodic function

I have the following question: The non-zero Fourier series coefficients of the below function will contain: The answer is: $a_0, b_n, n=1, 3, 5, \cdot \cdot \cdot$ So I first tried to ...
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34 views

Find assessment of residual member and prove it's convergence [on hold]

I want to find a relationship between $N$ and $ \varepsilon $. Evaluation of this expression is required: $$ |R_N(x,t)|= |\sum_{n = N+1}^\infty \frac{4\pi\mu_n sin(\mu_n) sin(\frac{\mu_n x}{l}) \...
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I was wondering whether or not these are equal to each other [on hold]

Does this $\frac{− cos (π n ) − 1}{πn}$ equal to $\frac{1}{nπ (1+(-1)^n+1)}$ ?
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26 views

analysis and fourier series [on hold]

Let $f$ a continuous fonction on $[-\pi,\pi]$, I would like to show : $\forall n\in \mathbb{N},\quad \left.\begin{array}{ll}\displaystyle \int_{-\pi}^{\pi}f(t)\cos (nt)\cdot dt=0\\\\\displaystyle \...
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74 views

Find the Fourier transform of $e^{- x^2}$.

I am trying to get the cosine Fourier transform of $e^{- x^2}$ where I am stuck in simplifying the integration further. The integration is between the limits $0$ to $\infty$!
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30 views

Suppose that $\hat{f}(n) = -\hat{f}(-n)\geq 0$ holds for all $n \in \mathbb{Z}$. Prove that $\sum^{\infty}_{n=1} \frac{\hat{f} (n)}{n} < \infty.$

Let $f$ be $1$-periodic and continuous. Suppose that $\hat{f}(n) = -\hat{f}(-n)\geq 0$ holds for all $n \in \mathbb{Z}$, where the $\hat{f}(n)$ denotes the $nth$ Fourier coefficient of $f$. Prove that ...
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Where does this derivation of the Fourier Series for csc(x) go wrong?

In this post, the following derivation for the Fourier series of csc(x) is given: \begin{align} \csc x &= \dfrac{1}{\sin x}\\ &= \dfrac{2i}{e^{ix}-e^{-ix}}\\ &= \dfrac{2ie^{-ix}}{1-e^{-...
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Showing that $-\frac{1}{2}=\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}$

So I've been trying to prove that $$\sum^{\infty}_{n=1}\frac{(-1)^{n}\sin(n)}{n}=-\frac{1}{2}$$ I've tried putting various bounds on it to see if I can "squeeze" out the result. Say something like (...
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Derivation of the Dirichlet Kernel in Fourier Analysis.

$$D_N(x)=\sum^{N}_{-N}e^{2\pi inx}=e^{-2\pi iNx}\sum^{2N}_{0}e^{2\pi inx}$$ $$=e^{2\pi iNx}\left(\frac{e^{2\pi i(2N+1)x}-1}{e^{2\pi ix}-1}\right)$$ $$=\frac{e^{2\pi i(N+1/2)}-e^{-2\pi i(N+1/2)x}}{e^{\...
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How do lie symmetries manfest in the metric tensors of their manifolds?

Suppose we are considering some pseudo-Riemannian manifold (spacetime) that has an underlying topology $SU(2)\times U(1)$ (which incidentally corresponds to a closed cyclic universe). Such a space is ...
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Weird Integral Involving Hermite Polynomials

I have stumbled upon the following integral involving the Hermite polynomials: $$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \...
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Real Part of the Dilogarithm

It is well known that $$\frac{x-\pi}{2}=-\sum_{k\geq 1}\frac{\sin{kx}}{k}\forall x\in(0,\tau),$$ which gives $$\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}=\sum_{k\geq 1}\frac{\cos(kx)}{k^2}.$$ ...
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1answer
11 views

Upper bound of a partial sum of Fourier coefficients

This problem is from Stein & Shakarchi's Fourier Analysis, Exercise 16(b) of Chapter 3. Let $f$ be a $2\pi$-periodic function which satisfies a Lipschitz condition with constant $K$; that is, $$|...
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Fourier Series of Product of Continuous Functions

Suppose $f, g$ are two continuous $1$-periodic functions. Prove that $$\widehat{f \cdot g}(n)=\sum_{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m),$$ where $\widehat{f\cdot g}$ is the Fourier coefficient of $...
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What is the Product of Fourier Coefficients?

Let $f, g$ be continuous $1$-periodic functions. Show that $$\widehat{fg}(n)=\sum_{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m).$$ Here the $\widehat{f\cdot g}$ denotes the Fourier coefficient of $f\cdot g$...
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Understanding a Fourier Series Problem

I found the following problem on Fourier Series. But I cannot make sense about how $f(x)$ could be a $1$-periodic function on the interval $[-1,1]$? Could you please help me make sense out of this ...
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Why a DFT of 2 senes is very noisy even with frequence sampling 5 times bigger?

I've set up this case to try to understand DFT implementing a real case in Excel Frame Size $\;\color{blue}{(T}$): 5 s Time Sampling $\;\color{blue}{(TS}$): 0,1 s Block Size $\;\color{blue}{(N = TT/...
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1answer
138 views

Fourier-Bessel Series for $f(x)=1-x$

I am trying to find the Fourier-Bessel series for the function $$f(x)=1-x\ \ \text{for} \ \ 0<x<1.$$ The Fourier-Bessel series has the form $$f(x)=\sum_{n=1}^{\infty} A_nJ_v(k_nx), \ \ 0<x&...
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Does The Fourier Series of $\sin(px)$ uniformly converges when $p \notin \Bbb Z$

As a part of a question that I solving about Fourier Series, I had to decide if the Fourier Series in the domain $[-\pi,\pi]$of $\sin(px)$ is uniformly convergent. Notice that $p>0$ and $p \notin \...
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Using Fourier Analysis to solve Basel Problem

Let $f$ be the $1$-periodic function such that $f(x)=x$ for $x \in [0, 1)$. Compute the Fourier coefficient $\hat{f}(n)$ for every $n\in \mathbb{Z}$ and use Parseval’s theorem to derive the formula $$\...
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Theta function equation

I have been trying to prove an equation of a theta function. I understood that it some how related to the Poisson summation formula, but no luck. Any help would be appreciated. the exercise
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1answer
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Find coefficient of fourier series knowing the series for a similar function

Knowing that $f(\theta)=\pi - \theta$ defined on $[0,2\pi]$ has the fourier series $\sum \limits _{n=1}^{\infty}\frac{2\sin(n\theta)}{n}$, then find the coefficients of $g(\theta)=\pi-\theta$ defined ...
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67 views

changes the sign of a continuous $f(x)$ with period 2$\pi$ at least 2n+2 times on a interval $[a,b]$ with $b-a>2\pi$

Let $f(x)$ is continuous on $\mathbb{R}$ with period $2\pi$, if there exists $n \geq 1, n \in \mathbb{N}$ we have \begin{equation} \int_0^{2\pi} f(x) dx = \int_0^{2\pi} f(x) \sin xdx= \int_0^{2\pi} f(...
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What is the relation between separation of variables and the eigenfunctions and eigenvalues for PDEs?

Studying Fourier Series and its application of solutions for Partial Differential Equations, in particular (historically) for the heat equation, one starts by separating variables. Somehow related to ...
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If $f$ is $2\pi$-periodic and $C^{k}$, $\hat{f}(n)=O\left(\frac{1}{|n|^{k}}\right)$ as $n\rightarrow\infty$

My question is: Suppose that $f$ is $2\pi$-periodic, and $C^{k}$. Show that $\hat{f}(n)=O\left(\frac{1}{|n|^{k}}\right)$ as $n\rightarrow\infty$. This notation means that there is a constant $C$ such ...
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“Bad” Fourier Series derivation

Let $f(\theta)$ $2\pi$-periodic such that $f(\theta)=e^{\theta}$ for $-\pi<0<\pi$, and $$e^{\theta}=\sum_{n=-\infty}^{\infty}c_{n}e^{in\theta}\,\,\, \mathrm{for}\,\, |\theta|<\pi $$ it's ...
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Proving that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi^{2}}{12}$

I want to prove that $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi^{2}}{12}, $$ using the Fourier Series for the $2\pi$-periodic function $f(\theta)=\theta^{2},\quad (-\pi<0<\pi)$, ...
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$L^1$-convergence

In my fourier analysis script we learnt that the fourier series of a function $f\in L^1[0,1]$ must not converge to $f$ in $L^1$, i.e. in general we have $||S_n(f)-f||_{L^1[0,1]}\not\to 0$. $S_n(f)$ ...
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Proving that $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{4n^{2}-1}=\frac{\pi -2}{4}$

I need to prove the equation below: $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{4n^{2}-1}=\frac{\pi -2}{4}$$ I think that I can use the following result provided by Fourier Series: $$|\sin\theta|=\frac{...
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Fourier Series integral proof

Consider a Fourier series representation of a function $f(x).$ Let $S_{N}(x)$ be the $N^{\text{th}}$ partial sum defined by $$S_{N}(x) = \frac{a_{0}}{2} + \sum_{n=1}^{N}a_{n}\cos\frac{n\pi x}{L} + ...
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proof that a fourier series converges to a pi periodic function

enter image description here so given the information on the left, how did they on the right hand conclude the first equality that $$S_{N}(f;x)-f(x) = \frac{1}{2pi}\int_{-\pi}^{\pi}g(t)sin(N+\frac{1}{...
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Has the fourier series for all smooth periodic functions finitely many terms?

When a fourier series has only finitely many terms, $f(x) := \sum_{n=-N}^N f_n e^{inx}$, then it is obvious that $f$ is smooth and periodic. Is the converse true aswell? If we have a smooth periodic ...
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Is mathematical relation between functions the same as relation between their Fourier series?

For example I have three functions $a(x)$,$b(x)$ and $c(x)$ which are defined on $(-\pi,\pi)$ They have a relation that $a(x)=2b(x)+c(x)$. Assume their Fourier series are $A(x)$ $B(x)$ and $C(x)$. ...
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1answer
128 views

Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?

I have the following problem: If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
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Looking for an explicit $f\in C(\mathbb{T})$ such that $t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{int}\notin L^\infty(\mathbb{T})$.

Denote the 1-torus with $\mathbb{T}$ and if $f\in L^1(\mathbb{T})$ denote the Fourier transform of $f$ by $\hat{f}$. I know that $$\exists f\in C(\mathbb{T}), t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{...
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2answers
25 views

Going from Fourier sum to Fourier integral - confusion on intermediate step

How does it solve the $A_0$ term? Can one prove this with an example? I don't know how to prove it to myself with an example In my lecture notes, my lecturer is trying to justify $C_n$ as weighting ...
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1answer
34 views

Fourier coefficients decay

This question popped up in a recent exam of mine to which I didn't know how to answer. The problem was as following Given the function $$f(\theta) = \frac{1+\cos(\theta)}{3+2\cos(\theta)}$$ and it'...
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86 views

Convergence of Fourier series in Bergman norm

Let $D$ be the unit disk, let $\nu>-1$, let $\varphi_\nu(w):=(1-|w|^2)^\nu$, let $\operatorname{d}\mu_\nu:=\varphi_\nu\operatorname{d}\mu$ where $\mu$ is the Lebesgue measure on the unit disk. If $...
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1answer
38 views

If the series of the Fourier coefficients is absolute convergent then the Fourier series is uniform convergent

It the following correct? If the series of the Fourier coefficients is absolute convergent then the Fourier series is uniform convergent. Is it related to some theorem? Parseval's identity?
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18 views

Poincare type inequality for functions with vanishing fourier coefficients

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Suppose $f\in L^1(\mathbb{T})$ and $f$ is also differentiable. Also assume that $f'\in L^p(\mathbb{T})$ for any $1\le p\le\infty$. Suppose that the Fourier ...
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1answer
36 views

Finding the value of $\sum_{n=1}^\infty \frac{1}{n^2}$ using Fourier transformation of a function f(x) [duplicate]

$f(x)=|x+\frac{\pi}{2}|-2|x|+|x-\frac{\pi}{2}|$ After using Fourier transformation I get $a_0 = \frac{\pi}{2}$ $a_n = \frac{8}{\pi n^2}sin^2(\frac{n\pi}{4})$ $f(x)= \frac{a_0}{2} + \sum_{n=1} ^{\...
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77 views

Cremona 2.14.1 Why is $c_4$ and $c_6$ complex when they should be rational?

In Cremona's online book Chapter 2, in order to calculate the lattice invariant we have: $\tau=\omega_1/\omega_2$ Set $q=e^{2\pi i\tau}$ (2.14.1) $c_4 =(2\pi/\omega_2)^4(1+240 \sum_{n=1}^{\infty} n^...
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Wavenumber $k$ becoming a continuous variable as $\lambda \to \infty$

I'm trying to piece together how to get to derive the Fourier transform from the Fourier series, reading my lecturer's notes. Now, if $f(x)$ happens to not be periodic, we can still give a ...
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32 views

Fourier Series Of $a^2-x^2$

Find Fourier coefficients of $ f(x)= \begin{cases} a^2-x^2, \mid x\mid<1 \\ 0, \mid x\mid\geq1\\ \end{cases} $ when $x\in (-\pi,\pi)$ What is the value at $x=1$ for which values of $a$ the series ...