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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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Plot the graph of the square wave function defined as $f(t)=\sum^{\infty}_{n=0}(-1)^n h_{n}(t)$ on the interval $t>0$ and find its Laplace transform.

Plot the graph of the square wave function defined as $$f(t)=\sum^{\infty}_{n=0}(-1)^n h_{n}(t)$$ on the interval $t>0$ and find its Laplace transform. The http://mathworld.wolfram.com/...
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How Does a Fourier $\sin$/$\cos$ Series Arise From a “Normal” Fourier Series? How Does This Relate to the Generalised Fourier Series?

I am told that Fourier showed that we can represent an arbitrary continuous function, $f(x)$, as a convergent series in the elementary trigonometric functions $$f(x) = \sum_{k = 0}^\infty a_k \cos(...
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Complex Fourier series of $ f=|2t|$ in domain $[-1,1]$

$ f(t)= \begin{cases} |t|&\text{if}\, -1 \leq t \leq 1\\ &\text{continue periodically} \end{cases} $ I need to find the complex Fourier series (so using the $f(t) = \sum\limits_{n=-\infty}^{...
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Find the Fourier series of the function: $f(x)=\begin{cases}x+1 & -1\leq x< 0,\\1-x &0\leq x< 1\end{cases},\;\;f(x+2)=f(x)$

I want to find the Fourier series of the function. For now, I am clueless on how to handle the function, int that it has $f(x+2)=f(x).$ $$f(x)=\begin{cases}x+1 & -1\leq x< 0,\\1-x &0\leq x&...
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Differential equation Fourier series solution, every even term should disappear but doesn't in my solution

I'm solving $$\frac{\partial u}{\partial t} - \frac{\partial^2u}{\partial x^2} = 1$$ $$u(0,t) = u(L,t) = 20, \quad L= 2.$$ $$u(x,0) = 20.$$ I start by homogenizing the equation. Let $u = v + \hat{u}, ...
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Fourier Sine series uses the orthogonal set $\{\sin(nx)\}^\infty_{n = 1}$ on $0 \le x \le \pi$

I came across the following explanation: The Fourier Sine series uses the orthogonal set $\{\sin(nx)\}^\infty_{n = 1}$ on $0 \le x \le \pi$. Here, the norm squared is $$|| \sin(nx) ||^2 =...
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Limit of simple Fourier series $\sum_{j=1}^{\infty}\frac{1}{j^k}\cos{\frac{2\pi j}{m}}$.

Are these limits known for all $m\ge1$? $$\sum_{j=1}^{\infty}\frac{1}{j^k}\cos{\frac{2\pi j}{m}}$$ $$\sum_{j=1}^{\infty}\frac{1}{j^k}\sin{\frac{2\pi j}{m}}$$
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+100

Coefficients in Generalised Fourier series Given By $c_k = \langle f, \psi_k \rangle$? $c_k$ Implicitly Chosen So That Sum Converges?

My notes state the following: The coefficients $c_k$ in the generalised Fourier series $$f(x) = \sum\limits_{k = 1}^\infty c_k \phi_k(x)$$ , with respect to the orthonormal set, $\{ \...
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Similar uses of Taylor Series and Fourier Series

Fourier series approximate given functions using sums of "sinx" curves with differing frequency. Taylor series approximate given functions using sums of power functions with differing degree. ...
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The $L^1$ convergence of Fourier series without absolute value

I am having trouble with the following qual problem. Assume that $f \in L^2[-\pi,\pi]$ and $c_{j}=\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)e^{ijx}\,dx$. (a) Prove that $$ \int_{a}^{b} f(x)\,dx = ...
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Constant term of weakly modular form of weight 2 vanishes

I stumbled upon this fascinating statement while browsing through old exercise sheets and don't find a fruitful approach to tackle the problem. Statement The constant term of the Fourier expansion ...
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If $f\in L^1(\mathbb{T})$ is such that $\forall n<0, \hat{f}(n)=0$, does the Fourier series of $f$ converges in $L^1(\mathbb{T})$ norm to $f$?

Let $\mathbb{T}$ be the 1-torus. Using the uniform boundedness principle, from the fact that $L^1(\mathbb{T})$ is a homogeneous Banach space and from $$\|D_N*\|_{1\rightarrow1}\ge\|D_N*F_n\|_{1}\...
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Weak convergence implies strong convergence in $L^1$ for Fourier series?

We say $\{f_n\}$ weakly converge to $f$ in $L^1[-π,π]$ if for each $g \in L^\infty[-π,π]$, $$\lim_{n\to\infty}\int_{-π}^{π}f_n(x)g(x)dx=\int_{-π}^{π}f(x)g(x)dx.$$ There is a question in my homework ...
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Using Riemann-Lebesgue lemma

I am reading a paper by VAALER. He is using Riemann-Lebesgue lemma and saying that below function tends to $0$ as $N \to \infty$ $$\int_{-2}^{2} \frac{\pi t}{\sin \pi t}(\cos \pi(2N+1)t)e^{2 \pi i t ...
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Evaluation of variants of Fourier coefficient

Suppose we know how to evaluate the following integral $$\int_0^{2\pi} f(x,\theta)e^{in\theta}\ d\theta=g(n, x).$$ Is there a general technique to evaluate/estimate $$\int_0^{2\pi} f(x,\theta)e^{i(h(\...
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Problem regarding Fourier Coefficients. [duplicate]

Question. Let $~f \in C^1[-\pi,\pi]$ such that $f(-\pi)=f(\pi)$ . Define, for $n\in \mathbb{N}$, $$b_n=\int_{-\pi}^{\pi}f(t) \sin nt~ dt$$ Which of the following statements are true? a. $b_n \to ...
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Heated rod problem with odd boundary

I am trying to solve the problem of a insulated heated rod given by $$u_t=ku_{xx}$$ With $$u(0,t)=0$$ $$u(x,0)=f(x)$$ and finally for $0<x<L$ at $x=L$ the rod gives off heat into a medium of ...
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Real Fourier series convergence

I have problem with determining function that the Fourier series is convergent to. My function is $$f(x) = \frac{\cos(1/x)}{(\ln(|x|/4))^2} $$ for $x\in [-\pi, \pi)\setminus \{0\}$ and $f(0) = 0$. ...
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Inverse Hausdorff Young

We all know the classic Hausdorff Young inequality for fourier series $\|\widehat{f}\|_{\ell^q}\leq\|f\|_{L^p}$, where $\frac{1}{p}+\frac{1}{q}=1$ and $p\in[1,2]$. My question is: is there any ...
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An interpolation inequality for Fourier cosine series

This question arises as a follow-up to MSE2874264. Let us assume to have $f\in L^2(-\pi,\pi)$ defined by $$ f(x) = \sum_{n\geq 1} c_n \cos(nx) $$ where the coefficients $c_n$ are non-negative, ...
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How to interpret this fourier basis function with Neumann boundary conditions on the rectangular domain?

So I was reading this research paper (http://geoggy.net/resources/Mathew10Metrics.pdf), came across this strange thing as Fourier Basis Function with Nuemann Boundary conditions on rectangular domain. ...
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Demonstration to check basis for fourier series

I have an issue about a question posted on another forum. The user gatsu on that forum posted (originally in French) that starting from the following hermitian inner product on periodic ...
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1answer
81 views

Sum of 1/n^4 using a half period cosine series

I am aware that I can solve the $$\sum_{n=1}^\infty\frac{1}{n^4},$$ using a a cosine series for $x^2$ on the half period $0<x<2$ however I am wondering if I can also solve this by using the ...
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If $f=0$ is the unique function for which the Fourier coefficients are zero then the set $\{\phi_{1}, \phi_{2},…\}$ is complete

Proposition: If the set of functions $\{ \phi_{1}, \phi_{2},... \} \ $ has the property that $ f = 0 $ is the only function with that all its Fourier Coefficients are zero, that is $C_{n}= 0\ $ $\...
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Using fourier series to get a summatory

I have the function $f(x) = e^{2x}$ in the interval $(0,2\pi]$ Using the formula $\int_0^{2\pi } e^{2x} e^{-inx}\, dx $ I get that $ {\mathbf{\gamma}}^{}_{n}= \frac1{2\pi} \frac{(e^{4\pi}-1)}{(2-...
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Derive Fourier transform by analogy to Fourier series?

The Fourier series coefficients are often derived by assuming a function can be represented as a series $$f(x) = \sum_{n=0}^\infty A_n \cos\left(\frac{2\pi n x}{L}\right) + \sum_{n=0}^\infty B_n \sin\...
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Proving Continuous Fourier Transform Formulas

Given a continuous non-periodic function, its Fourier transform is defined as: $$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(...
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Lissajous Curve dense in a Volume

The Lissajous-type curve $\left(\sin(\omega_1 t), \sin(\omega_2 t)\right)$, with $t \in \mathrm{R}$, is dense in a certain region of the plane. This can be seen for instance from excellent answers ...
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Calculation of sums using Fourier series [duplicate]

I will start my with an example, followed by questions based on this example, and then general questions that could apply to a general case. Calculate the sum : $\sum_{n=1}^\infty \frac 1 {n^6}$ The ...
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Filling gaps in “a proof” of Fourier inversion formula

Suppose $f\in L^1(\mathbb{R})$. Define the Fourier transform of $f$ as: $$\hat{f}:\mathbb{R}\rightarrow\mathbb{C}, \xi\mapsto\int_{\mathbb{R}}f(t)e^{-2\pi i\xi t}\operatorname{d}t.$$ Suppose that $\...
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Every set of orthogonal functions is complete?

Suppose that the functions $ \phi_{1}, \ \phi_{2},...,\ \phi_{n},...\ \ $ are orthogonal on the interval $[a,b]$. That is $$ \int_{a}^{b} \phi_{n}\phi_{m} dx = 0 \ \ \ \forall n\neq m$$ If $$ \...
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Every function can be represented as a Fourier Series - but why?

I can't really find a good answer to this question - the statement just seems to be assumed everywhere I look. Admittedly, I am not too well versed in the topic but I am an engineer and can understand ...
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Applying Bessel's inequality to a particular orthogonal system

I'm studying PDE with a book called "A first course in PDE with complex variables and transform methods", in this book the author presents Bessel's inequality as follows: $$ \sum_{n=1}^{\infty}{ C_{...
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Discrete Fourier Transform Matrix on different interval

I think I have a good understanding of the DFTM and fourier transforms in general. However, I'm running into some issues when I am approximating the Fourier series of a function on a general interval [...
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60 views

fourier series with random phases

Let $X$ be uniformly distributed on $[0,1]$ and consider the sequence of functions $f_n(X) = \sin(2n\pi X)$. Let $\phi:\Omega\rightarrow [0,2\pi]$ be a uniformly distributed random variable. Let $\...
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Given $a>0$, $\frac{1}{x^2+a^2}$ is not a Schwarz function.

Given $a>0$, $f(x) = \frac{1}{x^2+a^2}$ is not a Schwarz function. Please verify if this is correct: Although Poisson Sumation formula is working for this function $f$, I think it is not Schwarz,...
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Does piecewise continuity of $f'$ implies that the discontinuities of $f$ are jump discontinuities?

In this pdf, the following theorem is stated without proof: I'm not sure how this mathematically is accurate. My question is: Do the limits $f(x^+)$ and $f(x^-)$ exist, in the last statement? i.e. ...
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Fourier series - if it’s not an odd function, how does shifting affects my calculation?

The function $f(t)$ given in the graph for $-4 \le t < 4 $ this is the function of - $$f(t) = \begin{cases} -1, 0< t \le 1 \\ 0 , -1 < t \le 0 \end{cases} $$ I am told to determine the ...
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Projection-Slice Theorem for Fourier series

I understand the continuous version of the Fourier Slice-Projection theorem, which says that given a (nice enough) function $f:\mathbb{R}^2\to\mathbb{C}$ the following operations give the same result: ...
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Compute $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$ using Fourier series method.

I'm trying to prove the binomial identity $$\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}.$$ This is a problem from "Fourier Series and Integrals" by Mckean. So far, I have computed sums of the form $\...
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66 views

nth power of sine as sum of sine and cosine terms

I was working on integrals of the form $$\int_{0}^{\infty}e^{-x\cdot t}\sin^n(x) dx$$ and to solve them I tried to express $\sin^n(x) $ in form of a sum without any powers. Interesting for me I have ...
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How one can write the delta Dirac function using Fourier series expansion such that summation is performed over odd indices only?

The delta Dirac function can be presented in the form of Fourier series expansion as $$ \delta(x)=\frac{1}{2\pi} +\frac{1}{\pi} \sum_{n\ge 1} \cos (nx) \, . $$ The proof is straightforward and can be ...
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Fourier series: $\hat f(n)=O(1/n)$ and $f$ continuous implies uniform convergence?

Littlewood's Tauberian theorem: Let $a_n=O(1/n)$. (In particular, given any $0<r<1$, the power series $\sum a_nr^n$ converges.) If the function defined by the power series $$f(r)=\sum a_nr^n$$ ...
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how is this possible

I evaluated the Fourier series for $e^{-ax}$ , where $ -\pi \le x \le \pi$, which comes out be what is the first equation in picture tagged with this post below now what I am expected to do is to ...
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71 views

Differential equation with a Fourier Series

I have a piecewise function: $$ f(t) = \begin{cases} 7 & 0 < t < \pi \\ -7 & \pi < t < 2\pi \end{cases} $$ and it's assumed that when  f (t)  is extended to the ...
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21 views

Fourier cosine series for a piecewise function

I am trying to expand the following piecewise function as a cosine series: $$ f(x)= \begin{cases} 3 & -7 < x < -1 \\ 8 & -1\leq x\leq 1 \\ 3 & 1 \leq x < 7 \...
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47 views

CTFS of sine signal appended to dead signal

I have a signal which is a combination of sine signal and dead signal (0) as defined below. It is periodic with period 1. $$\begin{eqnarray} x(t) &=& sin(2*pi*1000*t) \enspace for \enspace 0&...
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2answers
49 views

Finding the Fourier series

I am trying to write the Fourier series for the following function $$f(x) = \left\{\begin{aligned} &0 && -\pi<x<0\\ &e^{-x} && 0<x<\pi \end{aligned} \right.$$ I ...
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1answer
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Fourier series for the function as defined below [closed]

$$ f(x) = \begin{cases} \pi & -\pi \leq x \leq \pi/2 \\ 0 & \pi/2< x < \pi\end{cases}$$ I tried following the definition to find the coefficients of the Fourier series for the above ...
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41 views

Problem on convergence of Cesaro mean of Fourier series

I'm working through Anton Deitmar's A First Course in Harmonic Analysis. And I got stuck at one of the problems: Using dominated convergence theorem, prove that $\lim_{n\to \infty} \sigma_n f(...