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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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Extra factor of 2 when evaluating an infinite sum using fourier series and parseval's theorem.

I'm asked to find the fourier series of the $2 \pi $ periodic function f(x) which is $sin(x)$ between $0$ and $\pi$ and $0$ between $\pi$ and $2\pi$ I use the complex form to proceed and get $$\frac{...
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Why the coefficients of G(t) depends on “n” in solving PDE wave equation by separation of variables

I don't understand why the coefficients of the sin and cos terms in G(t) (in the red box in picture 1) depends on "n" Why don't we simply choose them to be equal to 1? picture 1 picture 2 kreyszig'...
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Is there a continuous waveform that sounds the same as a square wave?

The fourier series $$f(t)=\sum_{n\in\mathbb N\\n\text{ odd}}\frac1n\,\sin(nt)$$ converges to a square wave. Square waves are discontinuous functions. I'm wondering if there's a continuous function ...
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recurrence relation for $\zeta(2n)$

I found this formula. Is it correct? For $n\in\Bbb N,\ n\geq2$, $$\zeta(2n)=\frac{2n\pi^{2n}}{\Gamma(2n+2)}+\sum_{k=0}^{n-2}(-1)^{k-n}\frac{\pi^{2n-2k-2}}{\Gamma(2n-2k)}\zeta(2k+2)$$ Here's my proof....
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Nonnegative Fourier series coefficients for periodic nonnegative-definite function

Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $\kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $\...
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how to compute the coefficients for half-range expansion of Fourier Series?

If a function $f$ is defined on $[-L, L]$, then its Fourier series is given by $$ \begin{aligned} a _ { 0 } & = \frac { 1 } { L } \int _ { - L } ^ { L } f \left( x \right) d x \\ a _ { n } & ...
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Fourier transform of $f(t)\cos(at)$ given fourier transform of $f(t)$

When the fourier transform of $f(t)$ is $F(w)$, how can I find the fourier transform of $f(t)\cos(at)$? Do I need to use some of fourier transform properties?
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Strict Inequality for Fourier Coefficients

I've been trying to solve this inequality but I only get the obvious part which is the $\leq$ part. I need the $<$. The problem is the following: Given a subset $A\subset [0,1)$ of measure not $0$,...
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$ \int_{0}^{1} \sum_{m=1}^{\infty} a_m \sin(m\pi x)\sin(l \pi x)dx = \frac{a_l}{2}$

Why is this true? $$\int_{0}^{1}u^0(x)\sin(l\pi x)dx = \int_{0}^{1}u(0,x)\sin(l\pi x)dx$$ $$ = \int_{0}^{1} \sum_{m=1}^{\infty} a_m \sin(m\pi x)sin(l\pi x)dx$$ $$ = \frac{a_l}{2}$$ where ...
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Doubt about Fourier Series, and its Uniform Convergence

Alright I have a question about Fourier Series. How does one prove that if a function is of Class $C1$ then the its fourier series in the interval $[-l,l]$ Converges Uniformly to the function $f$, ...
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Wave Equation and Fourier Series

I was given a guitar string of length 1 with fixed endpoints. My $f(x)$ is $2x$ if $(x \le 0.5)$ and $-2x+2$ if $(x \gt 0.5)$. My initial velocity is 0. $f(x)$ is the initial position I was first ...
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Generated space by the harmonic oscillators $e^{iw}$ in $[\pi , \pi)$ .

Let $\mathbb{T}$ represent the $[-\pi,)$ interval and for each $w\in\mathbb{Z}$ denote the function $$e_w:\mathbb{T}\to\mathbb{C}$$ such that $$\forall x\in\mathbb{T}, \ e_w(x)=e^{2\pi iwx}.$$ It is ...
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Fourier Analysis: Prove $c_n=a_nb_n$

So I am stuck in the middle of this proof: Let $f,g \in E[-\pi,\pi]$ be periodic functions with periods of $2\pi$, where $f(x)$ ~ $\sum_{n=-\inf}^{\inf}a_ne^{inx}$ $g(x)$ ~ $\sum_{n=-\inf}...
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Fourier series with complex coefficients

Hello i need to find the complex coffecients of the fouriers series of the $2\pi$ perriodic function: $$f(x)= \frac{\pi-x}{2}, \forall x\in]-\pi;\pi]$$ I stock in the computation of complex ...
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sum of the gray area using Fourier series [closed]

how to find sum of gray area at link above or sum of a series using Fourier series $s=\sum_{n=1}^\infty{An} $
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Find $\|f\|^2$ in two ways

Question: Consider the Hilbert space $L^2(\mathbb R/\mathbb Z)$ of periodic measurable functions. Recall that it has a Hilbert space basis consisting of functions $\chi_n(x)=e^{2\pi i n x}$ where ...
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How to calculate 8-point FFT of data by hand

Given the following data: Two period sine; samples = [0, 1, 0, -1, 0, 1, 0, -1]; I am asked to calculate the FFT of the sampled data to find the complex coefficients. I don't necessarily want the ...
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1answer
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Find Fourier coefficients of discrete odd signal

I'm self studying signal and system. I've come across this problem: if $a_1 = 1, a_2 = j$, what are $a_3, a_4, a_5$ for a discrete odd signal x[n] with fundamental period of N=6?
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Prove fourier coefficients of an odd discrete signal is $a_n = -a_{-n}$

I'm self studying signal and system. I've come across this property: fourier coefficients of an odd discrete signal is $a_n = -a_{-n}$, how can this be proved?
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Can $\int_{-\pi}^\pi e^{\cos(\theta)}\sin(\sin(\theta))\sin(nx)dx=\frac{\pi}{n!}$ be extened to non-integers?

It is well known that the fourier series for $e^{\cos(\theta)}\sin(\sin(\theta))$ is $\sum_{n=1}^\infty \frac{\sin(n\theta)}{n!}$ which implies that $$\int_{-\pi}^\pi e^{\cos(\theta)}\sin(\sin(\theta)...
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Constants for a solution to the Laplace equation

Consider one of the solutions to the Laplace equation $$u(x,t)=\sum_{n=1}^{\infty} C_{n}e^{-\alpha _{n}^2kt}\left [ \frac{-\alpha_{n} }{h}\cos(\alpha _{n}x) + B_{n}\sin(\alpha _{n}x) \right ],$$ ...
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Is it possible to construct a trigonometric series convergent in $(0,1)$ while divergent in $(2,3)$?

Here by a trigonometric series, I mean $$ f(x)=\sum_{n=1}^\infty a_n e^{ i b_n x }, $$ where $a_n$, $b_n$ can be arbitrary complex numbers. Two Questions: Q1. Is it possible to make such a ...
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Fourier series of Heaviside step function?

Let us say we have the Heaviside unit step function $\Theta(t-t^\prime)$. I want to calculate its Fourier series $$ \Theta(t-t^\prime)=\frac{1}{T}\sum_{n,m}\Theta_{\omega_n,\omega_m}e^{-i\omega_n t}e^{...
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Are the period of signal and its fourier coefficient the same?

Are the period of signal and its fourier coefficient the same?i mean if the the period of $x[n]$ is $5$,will the period of its fourier coefficient ,$a_k$, also $5$? how to prove it? Because the ...
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Finding the period of $x[n]$ when U only know its fourier coefficient

$a_k$ is the fourier coefficient of $x[n]$,and $a_k=1+2 \cos(\frac{k \pi}{4})+\cos(\frac{3k \pi}{4})$,how to find the period of $x[n]$ ? And fourier series is for the periodic signal, however, ...
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Use orthogonality to proof Parseval's identity for the general Fourier series written as the power spectrum

I need to show that $$\int_{-\pi}^{\pi}\left|\frac{a_0}{2}+\sum_{n=1}^{\infty}\alpha_n\cos(nx-\theta_n)\right|^2dx=2\pi\left(\frac{a_0^2}{4}+\frac12\sum_{n=1}^{\infty}\alpha_n^2\right)\tag{1}$$ Just ...
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Does $\cos\left(\frac{2\pi}2\right) = 0$?

I am studying chapter 10 (Partial Differential Equations and Fourier Series) of Boyce's Elementary Differential Equations and I stumbled upon this question: Consider the conduction of heat in a rod $...
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Sum of the series $\sum_{n\in \mathbb{Z}} \frac{e^{in x}}{(n+\alpha)^2}$

Here $\alpha $ is a non-integer real number. The series is apparently uniformly convergent and periodic. Is it possible to find a closed expression of it?
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Show ${\int_{0}^{1}|e^{-2\pi\cdot int} f(t)| dt} = {\int_{0}^{1}|f(t)| dt}$

The expressions below are from page 36 of Brad Osgood's "The Fourier Transform and its Applications". I don't understand how to establish the equality between the second and third expressions below (...
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Integral involving the log gamma function

I have used the Kummer representation series of loggamma function but does not look promissing to tackle this integral. Any idea to calculate this integral in closed-form ? $$\int_{0}^{1}\ln(x)\ln\...
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Fourier coefficients of a Holder continous function

I'm trying to do an exercise in Pinsky's "Introduction to Fourier analysis and wavelets": Suppose that $f$ satisfies $L^{2}$ Holder condition with $\alpha=1$. Prove that $\sum_{n\in \mathbb{Z}} |n|^...
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About Higher Order Fourier Analysis

This quastion is about the extension of Fourier analysis to higher orders. I know this is the approach to go beyond the linear phases. I am curious to understand it better. To be specific, Is it ...
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Aliasing - Nyquist frequency

According to Wolfram " in order to recover all Fourier components of a periodic waveform, it is necessary to use a sampling rate $\nu$ at least twice \textit{highest} waveform frequency. The Nyquist ...
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Uniform convergence of parametrised Fourier series.

Let $X$ be a compact topological space, and $f: S^1 \times X \rightarrow \mathbb{C}$ be a continuous function. Then for every point $x \in X$ we can compute the Fourier coefficients $$c_k(x) = \int_{S^...
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Level sets under heat flow

Let $f\in C^\infty([0,2\pi])$ be a smooth periodic function with mean zero ($\int f = 0$). Let $f(t,x)$ be the heat flow of $f(x)$, so that $f(t,x)$ solves $$ \partial_t f = f''. $$ Given a small $\...
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Why do choirs work?

In a choir, several people sing the same note at the same time. The sound each person makes consists of a tonic and certain overtones, but much of what we hear is the tonic frequency, so I'd like to ...
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Number of Fourier coefficients necessary to approximate $\delta(x)$ to given error

I am new to Fourier analysis, so I apologize if this question is very basic. For any integer $n$, I wish to construct a finite Fourier expansion $F_n$ with the following properties: $F_n(0) \geq 1$ $...
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1answer
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Pointwise convergence and Dirichlet's theorem on Fourier series

Let $f(x)=xcos(\alpha x)$ at $[-\pi,\pi]$ for $\lambda \in \mathbb{R}$. Let $S$ be $f(x)$'s corresponding Fourier Series. Prove or disprove: 1) if $\lambda \in \mathbb{Z}$ then $S$ Pointwise-...
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Period of $|\sin(\pi t)|$ - rectified wave, Fourier Series

I have a problem where I have to find the fundamental period and frequency of the following rectified periodic function $$ x(t) = |\sin(\pi t)| $$ From my understanding I know that the fundamental ...
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Difference in usage of power series and Fourier series [duplicate]

Both a power series and a Fourier series can be used to approximate a function. How do you use these differently?
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Is this function of bounded variation?

Consider Riemann's function defined on $\mathbb{R}$, $$ R(x) = \sum_{n=1}^\infty \frac{\sin n^2 x}{n^2} . $$ If you graph it, you can see that it shows a lot of zigzags. Hence, the question is, ...
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Help calculating Fourier series coefficients [closed]

I am trying to solve the following problem in my textbook. Use the Fourier series analysis equation (3.39) to calculate the coefficients $a_k$ for the continuous-time periodic signal $$ x\left(t\...
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Solving the heat equation gives constant solution

For the heat equation $$u_t = 4 u_{xx}$$ $$u(x,0) = 1$$ $$u_x(0,t)=u_x(1,t)$$ $$0 \leq x \leq 1$$ we know that the general solution for Neumann boundary conditions is $$u(x,t) = \frac{A_0}{2} + \...
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Sine series for $f(x) = x^2$

I'm asked to find the Fourier cosine and sine series for $x^2$. I know the steps. However, I'm conceptually confused because we learnt that even functions have only cosine terms, and odd functions ...
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Is the function in the $L^2$-sobolev space $H^{\frac{1}{2}} \ [- \pi, \pi]$ of order 1?

My professor and as a consequence the rest of my class are saying that the specified function is not in $L^2$-sobolev space $H^{\frac{1}{2}} \ [- \pi, \pi]$ of order 1. TLDR: Just skip ahead to $(\...
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No convolution Identity element in $L^1_{per} $ using Fourier series

We have to show that there is no identity element for the ring $ L^1_{per}( ]0,2\pi [) $ specifically using the Fourier coeffcients. Suppose that : $ \exists e , \: \: e * f = f \: \: \: \: \forall f ...
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Minimizing an Integral related to Fourier Series

Consider the inner product space $ C[0,2\pi] $ (or its completion, the Hilbert Space $ L^2[0,2\pi] $) as in 3.5.1 Find the values of $ c_1, c_2, $ and $ c_3 $ which minimize the value of $$ \int_0^{2\...
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G(k,X) is a modular form of weight k and character X

I'm trying to proof the transformation property of the Eisenstein series G(k,X) defined on page 17: https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf I already ...
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1answer
42 views

Discrete Fourier transform of exp(i k |m|)

Apologies if this is not mathematically very precise. I have been trying to calculate the Fourier series of $e^{i q |m|}$, but I'm having trouble with the absolute value in the exponential. Without ...
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Fourier-Legendre Series for $\arcsin{x}$

I have been trying to work out how to work out the coefficients in the Fourier-Legendre series of $\arcsin{x}$ (i.e., find $c_n$'s s.t. $\arcsin{x}=\sum_{n=0}^\infty c_n P_n(x)$ where $P_n(x)$ is the $...