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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Why do we have the following equality $\sigma(z)=\sum_{i=1}^n\lambda_if(w_i)e^{w_iz}$

Let $\mu=\sum_{i=1}^n \lambda_i\delta_{w_i}$, where $\delta$ is the Dirac measure, and $\lambda_i,w_i\in\mathbb{C}$ and $\lambda_i$ is non-zero. Let $\sigma_k$ be Fourier coefficients of $f$: $$ \...
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Is there an even function $f(k)$ such that the coefficients in its Fourier cosine series are the $3F2$ hypergeometric functions of the type below?

I have $f_{z}(k) = \frac{1}{2} a_0(z) + \sum\limits_{n=1}^{\infty} a_n(z) \cos n k$. where $a_n(z) = {}_3F_2(\{1/2, 1, 1\}, \{ 1-n, 1+n \}, z)$. Is it possible to find $f(k)$ explicitly?
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Finding $MSE$ of optimal estimator

Background of the question: We know that $X$ is a continuous random variable that has $P\left(-1\le X\le 1\right)=1$ and $f_X\left(x\right)<\infty $ We define $X(n)=cos(\pi n X)$ $E[X]=\mu$ and $...
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Problem with one implication

Let p$\in \mathbb{N} ^ *$and $g \in C^{p } (\mathbb{R} , \mathbb{C}) $2-periodic function. Prove that $c_{3k}(g) = 0 $(its fourier coefficient)for all $k \in \mathbb{Z}$ if and only if there exists ...
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Jacobi-Anger expansion of Fourier series

I know well that the Jacobi-Anger expansion is $e^{iz\sin (x)}=\sum_{n=-\infty}^\infty J_n(z)e^{inx}$ or $e^{iz\cos (x)}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{inx}$ My question is how to expand with ...
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In a fourier series if $a_0$ is equal to 0 is $a_n$ also equal to 0? [closed]

When im trying to determine a fourier series if I determin the $a_0$ is 0 does it follow that $a_n$ must also be 0?
Danilo 357's user avatar
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What is the limit (as a distribution) of this Fourier series, similar to that of a Dirac Delta?

I know that a representation of the Dirac delta function is $\sum_{n=-\infty}^\infty e^{inx}=2 \pi \delta(x)$. I am trying to figure out if the series with positive $n$ only $\sum_{n=0}^\infty e^{inx}$...
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Dirichlet Initial-boundary-value problem

I'm currently solving an initial-boundary-value problem with Dirichlet boundary values, and are stuck on one part that I can't say what is wrong and right. I got the following PDE $$\begin{cases}\...
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Can $\int_0^\infty \frac{\sin x}{x}dx$ be evaluated using Fourier series?

I'm aware of the many answers to evaluating the integral of $\frac{\sin x}{x}$ over $[0,\infty)$ given here. No answer seems to mention the approach below, and neither have I found anything elsewhere, ...
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Exception where $L^2$-Holder condition does not hold

Recently, I have tried to resolve the following problem. Let periodic $f$ be defined by the following Fourier series, \begin{align} f(x) = \sum_{k=1}^\infty \frac{1}{k^{3/2}} e^{ikx}. \end{align} ...
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Investigate convergence for both series on the interval $(0,1)$.

I am given a function $f(x)=x^2$ on $(0,1)$. First, I found the cosine series for the even periodic extension on $[-1,1)$ which is $\frac{3}{2}+\sum_{n=1}^{\infty}\frac{4(-1)^n}{(n\pi)^2}cos(n\pi x)$. ...
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Use the result* and Parseval's identity to compute the value of $\sum_{n=1}^{\infty} \frac{1}{n^2}$

The result* is the fourier series for $f(x)=x+\pi$ on $(-\pi,\pi)$ is $\pi+\sum_{n=1}^{\infty} \frac{-2(-1)^n}{n} sin(nx)$ I know Parseval's identity derived from the fourier series of a function is $\...
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basis for $\mathcal M_{24}$

When I run the command ModularForms(1, 24).basis() in Sage, I get back the following $q$-expansions: $q + 195660q^3 + 12080128q^4 + 44656110q^5 + O(q^6)$ $q^2 - 48q^...
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Exploring the Connection Between Odd-Indexed Fourier Coefficients and Transformed Signals

When we delve into signal processing, we encounter the concept of Fourier series, which enables us to represent a signal (s(t)) through its Fourier coefficients (a_m). These coefficients are pivotal ...
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Fourier sine series sum for odd values of n

I'm working through Griffiths electrodynamics, and there was a Fourier sine series that I couldn't figure out how to explicitly sum - the infinite sum for odd values of n of $$\frac{1}{n}e^{-\frac{nπx}...
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Holder condition on $\sum_{k=0}^\infty a^k \cos(b^k x)$

I'm struggling with solving the following problem. Let $0<a<1$ and $2\leq b \in \mathbb{Z}$, so $b$ is an integer. Prove that the following series \begin{align} \sum_{k=0}^\infty a^k \cos(b^k x) ...
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Expansion of Gaussian wavepacket in terms of sines

From this paper, I am trying to compute the coefficients in the expansion of the Gaussian wavepacket $$\phi(x) = \frac{1}{(2\pi\sigma^2)^{\frac{1}{4}}}\exp \Big(-\frac{(x-x_{0})^{2}}{4\sigma^{2}} + ...
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Fourier transform on a regular lattice: The restricted set of wave-vectors

I) Consider a function defined only on the vertices of a 1D regular lattice: $f_i \equiv f(x_i)$ for all $x_i$, $i \in \{ 1, 2, ..., N \}$ and $x_{i+1} - x_i = a$, where $a > 0$ is the “lattice ...
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What is the formula for a0 in the fourier series coefficient and what is the significance of the term before? [duplicate]

I'm trying to understand the fourier series coefficients for trig fourier series, and I came across the general expansion written in this form. And their respective formulas, and I found an ...
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Can a gaussian be expressed as a sum of sines or cosines? [closed]

I'm taking a quantum mechanics course in university right now and we're dealing with gaussian wave packets and I'm particularly interested in whether a gaussian can be expressed as a sum of sines or ...
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On Parseval's identity and computing a special sum

In Vretblad's Fourier Analysis and its Applications, I've encountered the following sum Prove the formula $$\sum _{k=0}^{\infty }\frac{\left(-1\right)^k}{\left(2k+1\right)\left(\left(2k+1\right)^2-\...
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Katsnelson Harmonic Analysis chapter 5 exercise 5

I tried many ways to solve this problem but none seems to be conclusive :( any help will be great. Suppose $f\in L^1(\mathbb{T})$ with Fourier coefficients $\hat{f}(n)=O(|n|^{-k})$ where $k$ isn't ...
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Fourier series for $f(x) = \cos(3x)$

I tried to to fourier series for $f(x) = \cos(3x)$, and I keep getting $0$ for all coeficients, $a_0$ and $a_n$. Am I missing something? Could $\cos(3x)$ be fourier series on its own?
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Definition of Besov space $B^\alpha_{2,\theta}(-\pi,\pi)$ and embeddings

I consider Besov space $B^\alpha_{2,\theta}(-\pi,\pi)$ defined as follows for $2\pi$-periodic functions, \begin{align} B^\alpha_{2,\theta}(-\pi,\pi)=\left\{f~\Bigg|~\sum_{j=0}^{\infty}\left(\sum_{2^j\...
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Characterization of non-negative Fourier series

Let $f(\omega) = \sum_{k = 0}^{n-1} w_k \cos(k \omega)$ with $\omega \in [-\pi, \pi]$ be a finite and real Fourier series. Is it possible to characterize for which coefficients $w_k \in \mathbb{R}$ ...
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Finding periodic solutions of a second degree differential equations

Let $w\in \mathbb R$ and $p>0$. I want to find the p-periodic solutions of the differential equation $$ f''(x)+w^2f(x)=0. $$ Let the $p$-periodic solution be $$ f(x)=c_0+\sum_{k \in \mathbb Z \...
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Quickly finding/eliminating coefficients of the complex exponential form of the Fourier series

A typical question that is found on an (MCQ) exam for a course I'm taking is to express a periodic function in the complex exponential form of the Fourier series $x(t)=\sum\limits_{k=-\infty}^\infty ...
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Calculating Fourier coefficient of a split function with time period $y\left(x\right)=\begin{cases}-1&-T<x\le 0\\ 1&0<x\le T\end{cases}$

$$y\left(x\right)=\begin{cases}-1&-T<x\le 0\\ 1&0<x\le T\end{cases}$$ I tried to use definition: $B_k=\frac{1}{2T}\int _{-T}^{+T}y\left(x\right)e^{-jk\omega _0x}dx\:\:=\frac{1}{2T}\int _{...
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Eigenvalue problem for linearized reaction-diffusion system

I recently started studying about reaction-diffusion system and Hopf bifurcation theory. I realize that Fourier transform/series is a quite useful tool here but it's been many years since I got ...
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what does it mean that Fourier series applies to a periodic function?

I'm very new to Fourier series (as well as to math stack exchange), and have a very basic level of confusion. Suppose I have a real-valued function $f(t)$ defined in $t \in [0, T]$ (for some fixed $T$)...
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Question about heat equation and partial derivatives equations

Consider the problem: $u_{t}=Ku_{xx}$ $u_{x}(0,t)=g_{0}(t)$ $u_{x}(L,t)=g_{1}(t)$ The question is "Transform this problem in other where the frontier conditions are homogeneous, suposing $g_{0}$...
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Fourier coefficients (full-range series)

For the periodic function $f(\theta)=(\theta^2-\pi^2)^2$ on the interval $-\pi\leq\theta<\pi$, show that it has the Fourier series $$f(\theta)=\frac{8\pi^4}{15}+24\sum_{n\neq0}\frac{(-1)^{n+1}}{n^4}...
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Discrete Fourier transform of repeating values

In this post, I call $\hat{v}\in\mathbb{C}^N$ the Discrete Fourier Transform of $v\in\mathbb{C}^N$ the vector such that: $$ \hat{v}_j = \sum_{k=0}^{N-1} v_k \exp\left(-2i\pi \frac{kj}{N}\right) $$ For ...
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From Fourier series to DFT

A complex function $f(x)$ that is periodic on $[0,2L]$ can be represented as infinite sum of a complex, orthonormal exponential functions that represent the frequencies that reside in $f$. $$ f(x) = \...
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Representing any even function as a sum of cosines

I'm looking into the Fourier series, and I understand that any function can be written as a sum of an odd and even function. However, I don't understand why any even function can be written as sum of ...
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Expand non-trivial function in Fourier series

I urgently need help on the following Fourier series expansion: $f(x) = -1$ on $(-\pi, 0)$ and $f(x) = x$ on $(0, \pi)$. I would like to find the general formula for the Fourier coefficients, but I ...
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How can this related to the Dirichlet 's Kernel

The problem is that: Show that for all $a \in \mathbb{R}$, $$|\sin(ax)\sin\left((a+\tfrac12)x\right)|\geq |\sin^2(ax)\cos(\tfrac{x}{2})|-|\sin(\tfrac{x}{2})|.$$ My idea is through simple calculation ...
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Fourier series zero coefficients of derivative of a function

Based on Openheim's Signal and Systems book part 3.5 and with the same logic in here. We can conclude that if: $$ f(x) \overset{FS}{\longleftrightarrow} a_k $$ then: $$ \frac{d}{dx} f(x) \overset{FS}{...
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Fourier series for a function with a "jump point".

Find the Fourier' series of $f(t)$, which: $$ f(t)= \begin{cases} \pi^2-t^2 & \text{if} & t\neq 1/\pi^n & n\in \mathbb{N} \\ t^2 &\text{if} & t= 1/\pi^n & n\in \mathbb{N} \...
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Construct a manufactured solution of Poisson's equation with Chebyshev/Fourier expansions

I am solving a nonlinear Poisson's equation numerically using a mixed Chebyshev/Fourier spectral methods. Thus, assuming $x$ is periodic and $y$ is nonperiodic. I am trying to test my current ...
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On differentiability and Fourier coefficients; Vretblad

Background I'm reading Vretblad's Fourier Analysis and its Applications. In the chapter on Fourier series, there is a section on differentiable functions. Let $\mathbb T$ be the unit circle and denote ...
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Decay rate of $\vert x \vert^\alpha$ is $\mathcal{O}(\vert n \vert^{-1-\alpha})$

Now I'm really stuck on proving the following. Let $f(x)=\vert x \vert^\alpha$, $x\in[-\pi,\pi]$, $0<\alpha<1$. Show that for $0<c_1<c_2$, as $n\rightarrow\infty$, $$ c_1\vert n \vert^{-1-\...
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Estimating L1 norm of a trigonometric polynomial related to Dirichlet kernel

I am trying to estimate the norm of an L1-multiplier. So, I would like to find an upper bound for the L1 norm of the following trigonometric polynomial: $$h(x)=\sum_{k=b}^{k=4b}\frac{k^{\alpha}}{\log^{...
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Form of fourier series in 3D coordinate

I have an interesting problem where a constrained solver is used to estimate a periodic function as part of a sensor calibration process. The idea is to write this periodic function in the form of ...
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Relationship between Fourier inversion theorem and convergence of "nested Fourier series representations" of $f(x)$

In this question I use the term "nested Fourier series representation" to refer to an infinite series of one or more Fourier series versus a single Fourier series. Whereas a single Fourier ...
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Pole expansion and Fourier series of lemniscate sine function

Given that $$\frac{\varpi}{\text{sl}(\varpi z)}=\sum_{n,k\in\mathbb{Z}}\frac{(-1)^{n+k}}{z+n+ik}$$ It can be deduced that, for $-1<\text{Im}(z)<1$: $$\frac{1}{\text{sl}(\varpi z)}=\frac{\pi}{\...
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Third-order PDE and Fourier series

I am wondering how to find a Fourier series solution of the following PDE: $$u_{t} + i u_{xxx} = 0$$ $$u(0, x) = u_{0}(x)$$ where $t$ is in $(0, \infty)$ and $x$ is in the torus $\mathbb{T} = \mathbb{...
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Question about derivation of Fourier transform

I'm reading the book "Fourier Analysis and Its Applications" and in the derivation of the Fourier transform he began with writing the Fourier series $$f(t)=\sum_{n=-\infty }^{\infty }c_{n}...
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The correct way of looking at Fourier transform

I Know that Fourier transform states that any non-periodic function could be described as summation of sines and cosines by saying that $$F(w)=\int_{-\infty }^{\infty }f(x)e^{^{-iwt}}dt$$ And this was ...
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Can you derive a closed form for this Fourier series: $\sum_{n=1}^{\infty} \frac{1}{n^2}\frac{a^n+b^n}{1-(ab)^n}\sin(n\phi)$

During solving Laplace equation on an annular domain using separation of variables I encountered the following Fourier series: $$\sum_{n=1}^{\infty} \frac{1}{n^2}\frac{a^n+b^n}{1-(ab)^n}\sin(n\phi)$$ ...
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