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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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Finding out the number of minima for a fourier expansion

Suppose I have a Fourier series f(x) = $\sum_{n=1}^N t_n \cos(nx)$ defined in the domain $(-\pi,\pi]$. we need to prove that mathematically we can ' at most' have N minima points excluding the ...
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Is there a compact form of $\frac{2}{n+1}\sum_{k=1}^n\frac{\sin^2(\pi jk/(n+1))}{z-2\cos(\pi k/(n+1))}$ in terms of $z$, $n$, $j$?

I would like to evaluate the following sum: $$S_j(z,n)=\frac{2}{n+1}\sum_{k=1}^n\frac{\sin ^2\left(\frac{\pi j k}{n+1}\right)}{z-2 \cos \left(\frac{\pi k}{n+1}\right)}$$ where $z\in\mathbb{C}$ is an ...
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$L^2$ vs $L^\infty$ projection

Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is $$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$ where $d_\...
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Fourier coefficients for $f(x) = 3 - \sin(3x)-\dfrac{1}{3}\cos(9x)$.

I am trying to find the fourier coefficients for $f(x) = 3 - \sin(3x)-\dfrac{1}{3}\cos(9x)$ I have understood that the overall period of the function is $\dfrac{2}{3}\pi$, and can due to formulas find ...
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Recovering Fourier series coefficients from the Fourier transform of a function extended on the unit circle.

I'm working on a problem involving Fourier transforms and functions extended on the unit circle. Given a function $f(x)$, I'm considering its extension on the unit circle and its Fourier transform. ...
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I the list of Fourier coefficients unique for every function?

I’ve recently encountered in my studies a way to express functions as vectors, in particular using the Fourier coefficients and the complex exponentials as a base. This means that $\vec{c} \cdot \vec{...
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Rate of Uniform Convergence of Fourier Series to a Smooth Function?

I'm wondering if there are any known results on the rate of uniform convergence of a Fourier partial sum to a smooth function ?. More specifically, I am wondering ...
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FFT for the Estimation of Power Spectra (Welch's Method) - DFT Definition

I was reading Peter Welch's famous paper "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Aver. aging Over Short, Modified Periodograms" the ...
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If we know the even fourier series are we able to find the odd version?

If we have a sequence, call it $a_{n}$ where $n>0$, and we take it as the fourier coeficents of an even function... $$ F_{e}(x) = \sum^{\infty}_{n=1}a_{n}\cos(2\pi nx) $$ ... and we know the form ...
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Difficulty in proof of a lemma in Katznelson's book about Harmonic Analysis chapt. 2 section 3 (divergence sets)

To explain my problem I must insert more from Katznelson's book than the part where I have a difficulty. (My comments to these copies in red.) Beginning of book quote End of book quote In the remark ...
Ulysse Keller's user avatar
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Series expansion involving integrals of higher derivatives and Bernoulli polynomials

Given a (smooth) function f defined on $[0, 1]$, I am looking for a series expansion of the form $$ f(x) = \sum_{n = 0}^\infty \frac{c_n}{n!} P_n(x), $$ where $c_n = \int_{0}^{1} f^{(n)}(t) dt$, where ...
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Nonhomogeneus PDE function requires expansion in sines?

I'm studying about the solution to the PDE: $$ \Delta u(x,y)=-f(x,y)\\ u(0,y)=u(a,y)=0 \\ u(x,0)=g(x) \ \ , \ \ u(x,b)=h(x) $$ And the first step is to start solving it like a homogeneus equation with ...
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How to find best fitting trend that fits this periodic data?

I have this data only for this month and I need to find a good trend function. I used a polynomial of degree 6 but with that you can't really extrapolate. I heard you should use Fourier series? It is ...
Chase Bluehorn's user avatar
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Is there uncertainty principle for Fourier series?

I know there exists many types of uncertainty principle for Fourier transform. I tried to search but I couldn't find any such principle for Fourier series
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Weighted inequality on torus

In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
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Finding a symbol sequence such that the PSD is 0 at certain frequencies

Let $\left\{ C_{n}\right\} $ be an arbitrary sequence of real numbers(symbols). Lets denote the Power Spectral Density with some shape filter such that. $$ \tilde{S}_{d}\left(f\right)=\frac{T}{4}sinc^...
Danny Blozrov's user avatar
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How to prove that sum of infinite complex harmonics makes a continuous time impulse function?

I was trying to substitute complex fourier series coefficients into complex fourier series formula and I came up with this identity in order to get the same function back. How can I prove that this ...
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Plot of differentiated Fourier series

I have the following task: draw a plot of the sum of differentiated Fourier Series for $f(x) = e^{-|x|}$ for $x \in [-\pi, \pi]$. Looking at the plot of Fourier series we can say that all coefs with $...
John Doe's user avatar
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Finding $f$ orthogonal to set.

Let $A \subsetneq \mathbb{Z}$. Consider the following set of functions $$M = \{e^{2 \pi in x}\}_{n \in A} \cup \{xe^{2 \pi i n x}\}_{n \in A^c}$$ belonging to $L^2[0, 1]$. Does there exists a function ...
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Convergence of Fourier Series in Mean Square

I have a question regarding the convergence of the Fourier series of a function. I am studying Fourier analysis from M.Stein and in Chapter 3 it states the theorem that if a function $f(x)$ is ...
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Diverging Fourier Series of Sawtooth Function

I am studying Fourier Analysis from the M.Stein book, in Chapter 3 there is a whole few pages dedicated to showing that the Fourier series of the sawtooth function defined as $$f(x)=i(\pi -x)$$ odd in ...
Riccardo Caiulo's user avatar
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Jacobi-Anger expansion for non-integer power

One way of writing the Jacobi-Anger expansion is $$ \frac{1}{2\pi} \int_0^{2\pi} e^{i (n \theta - a \sin \theta)} d \theta = J_n(a) $$ for real $a$ and integer $n$. Is there a corresponding formula ...
Bio's user avatar
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How is the following Matsubara sum solved? $\sum_{n} \frac{2x}{2i(2b - i\omega_{n})^{2} + x^2}$

$\sum_{n} \frac{2x}{2i(2b - i\omega_{n})^{2} + x^2}$, where $b$ represents some chemical potential and $\omega_{n}$ are the bosonic frequencies. The conflict for me is the factor $2i$ that appears in ...
Ricardo Löwe's user avatar
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If $N$ is odd, is $\sum_{k=1}^{(N-1)/2}\frac{1}{\sin^2(\pi k/N)}$ always rational? [duplicate]

For an odd number of $N$, derive the following $$\sum_{k=1}^{(N-1)/2}\frac{1}{\sin^2(\pi k/N)}$$ Numerically, it seems to be a rational number, but I can't prove it. For $N=3$, the answer is $4/3$. ...
Yoshiki S's user avatar
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Why is this set of functions compact in L^1 (in proof of localization principle by Katznelson)

These are the hypotheses of the localization principle, a theorem in Katznelson's Introduction to Harmonic Analysis (to be found in chapt.2, section 2): Let $f$ be a complex-valued periodic function ...
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Uniform convergence of Fourier expansion of a 3rd-differentiable function

Let $f$ be a 3rd-differentiable function defined on $\mathbb{R}$ such that $f(x+2\pi)=f(x)$ for all $x\in\mathbb{R}$. Suppose that the Fourier expansion $\frac{a_0}{2}+\sum\limits_{n=1}^{+\infty}(a_n\...
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Difference between formula DFT (Discrete Fourier Transform) and trigonometric interpolation

Assume we have a discrete-real-valued scalar function $f[n]$, which is sampled equally spaced with distance $\Delta n$ using $M=2m-1$ samples. $f[n]$ is also periodic with respect to $2\pi$. My goal ...
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How to find a Fourier series using trig identities

How to find the Fourier series of $$f(x) = \sin^3 (x) - \cos^3 (x) + \cos^4 (x)$$ My professor said we won't need to solve integrals but I am not sure why ? I am confused why we won't find the Fourier ...
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Finding the Fourier Coefficients of a repeated non-periodic signal

I need some help understanding the steps behind this exercise. Find the Fourier coefficients of $x(t) = rep_{10}e^{-t}u(t)$ where $u(t)$ is the unit step signal. I know the alternate way of writing ...
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Rapidly decaying Fourier coefficients for $\sum e^{-(x-2n)^2}$ [duplicate]

Consider the function $$f(x) = \sum \limits_{n \in \mathbb{Z}}e^{-(x-2n)^2}$$ This is an even periodic function with period 2, so its Fourier series has only cosines and a constant. Calculating this ...
Ron Shvartsman's user avatar
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Deriving the Fourier Transform from the Fourier Series

Consider a function $f(x)$ with support on $I = [-L/2, L/2]$. We can expand $f$ into a Fourier series over the interval $I$: $$ f(x) = \sum_{n=-\infty}^{\infty} a_n e^{2 \pi i n x / L}, $$ where the ...
Mark's user avatar
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Laplace's Equation on a Pac-Man

I am struggling way too much with this problem, any help is highly appreciated. Consider the Pac-Man-like set described by $$P=\left\{(\rho\cos\theta,\rho\sin\theta):\rho\in(0,1),\theta\in\left(\frac{\...
tripaloski's user avatar
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(Infinite) sum of products of 'offset' Bessel functions (an application to calculating the RMS power of a harmonic "FM" oscillator)

TLDR version: I'm trying to calculate $\sum_{n=1}^{\infty} J_{n-1}(a) J_{-n-1}(a)$ but can't seem to get it right. For motivation & attempt, read the rest. Consider the harmonic "FM" ...
got trolled too much this week's user avatar
2 votes
1 answer
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Representation for real part of a holomorphic function on the circle

Let $f$ be a holomorphic function on the closed unit disk $\{z: |z|\leq 1\}$, with $u$ its real part and $v$ its imaginary part, i.e. $f = u + i v$. Suppose $u$ fulfills that $$ \int_0^{2\pi} u(1, \...
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1 answer
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Evaluating the integral in an integral representation of the sum that is the Fourier series for $e^{ax}$

I aim to evaluate the following sum through an integral representation of it: \begin{align*} & \frac{\sinh a \pi}{\pi} \sum_{n=-\infty}^\infty \dfrac{(-1)^n}{a - i n} e^{i nx} \nonumber \\ & = ...
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Compute $\int_{\phi_1}^{\phi_2} \left( \sum_{n=0}^N a_n \cos(n \phi) \right)^2 \,\text{d}\phi$ analytically?

For a computational model I need to evaluate the following integral many times: $$ I(\phi_1, \phi_2) = \int_{\phi_1}^{\phi_2} \left( \sum_{n=0}^N a_n \cos(n \phi) \right)^2 \,\text{d}\phi, $$ where I ...
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3 votes
1 answer
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Properties of fourier series on $SO(3)$

With standard fourier series we can use some identities like convolution theorem and Parseval's theorem: (convolution theorem) Fourier series of the convolution of $f$ and $g$ is the point-wise ...
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Real Valued 2D fourier series over a complex domain (and 2D manifold)?

This answer does a great job of explaining the fourier basis for $\mathbb{R}^2$, however, the answer assumes a square domain. Is it possible to generalise the answer to mor complicated domains? Like a ...
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Integral of complex exponential with phase given by multiplication by a rational

I want to check that if $f \in L^1(\mathbb{T})$, $m$ is a positive integer, and $f_m(t) = f(mt)$ then $$\widehat{f_m}(n) = \begin{cases}\hat{f}(n/m) &\text{ if } m \mid n, \\ 0 & \text{ if } m ...
approximate-identity's user avatar
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Linking Fourier Coefficients of periodic functions

Let $\tau\in (0,1)$ and assume that we have a $\tau$-periodic function $$f_1(t) = \sum\limits_{k\in\mathbb{Z}} a^{(1)}_k e^{\frac{2\pi i k}{\tau}t},$$ a $(1-\tau)$-periodic function $$f_2(t) = \sum\...
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How to obtain the result of Ott-Antonsen ansatz of the classic Kuramoto model

This is my derivation process: $\frac{\partial f}{\partial t} +\frac{\partial (f*\frac{\partial \theta }{\partial t})} {\partial \theta } = 0.$ Then $\frac{\partial}{\partial \theta }\left(f \cdot \...
Putin's user avatar
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2 votes
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Decomposition of function into products

Given a single variable function $f(x)$, is there a way of decomposing it into the product of a family of function. Something similar to, $$f(x) = \prod_n p^{a_n}_n(x)$$ I am trying to find the ...
PRITIPRIYA DASBEHERA's user avatar
1 vote
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Fourier series related to the floor function

I'm trying to find the Fourier series for the periodic function: $$f(x)=\left \lfloor nx\right \rfloor-n\left \lfloor x\right \rfloor\;\;\;\;\;n\in \mathbb{N}$$ $\left \lfloor \cdot \right \rfloor$ ...
Mohammad Al Jamal's user avatar
3 votes
1 answer
54 views

Sufficient conditions on Fourier coefficients that imply smoothness

Given a square-integrable function $f$, let $\hat{f}(n)$ denote its Fourier coefficients. It is well-known that differentiability of $f$ implies that the $\hat{f}(n)$ satisfy certain bounds, i.e. they ...
student566's user avatar
1 vote
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Decay of Fourier coefficients of a function on the disk

What can be said about the decay of the Fourier coefficients of a smooth function on the unit disk $B_1(0)$? More precisely, I'm in the following setting: Let $v \in \mathcal{C}^\infty(B_1(0), \mathbb{...
mathology's user avatar
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Show $\frac{1}{\pi^n}\zeta(n) \in \mathbb{Q}$ for even n

I am working on an exercise with aim to prove $\frac{1}{\pi^n}\zeta(n) \in \mathbb{Q}$ using Fourier series. Here's the outline: We first let $f_1(x) = x-\frac{1}{2}$, then we can express it with ...
aawangas's user avatar
1 vote
1 answer
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Complex exponential Fourier coefficients of a convolution involving the exponetial function

In the book "Elementary Classical Analysis", by Marsden, the following is proposed as a worked example: Let $f:[0,2\pi]\to\mathbb{R},g:[0,2\pi]\to\mathbb{R}$ and extend by periodicity. ...
Pablo Álvarez's user avatar
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Consequence of Fejer's theorem

I'm reading calculus conspect of my university and that's how Fejer's theorem is stated: let $f \in C(T)$ is a continous, $2\pi$ periodic function, then $\sigma_N(f) \xrightarrow{\xrightarrow{n}} f$ ...
myfakeaccount's user avatar
1 vote
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When I was looking at the proof of Fejer's theorem, I encountered a problem in the derivation of a formula. [closed]

How did we get the last equation? Why can the summation be converted into a square term? $$ \begin{align}K_m(x)&:=1+\frac{2}{m}\sum_{j=1}^{m-1}(m-j)\cos(jx) \\&= \frac1m\sum_{j=-(m-1)}^{m-1} (...
mse xing's user avatar
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42 views

On the convergence of $\sum_{n\geq0}a_n\cos(nx)$

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers such that $\sum_{n\geq0}a_n$ converges absolutely. Can we say that $\sum_{n\geq0}a_n\cos(nx)$ converges to a continuous function? I know ...
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