Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

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1answer
27 views

Problem understanding half wave symmetry

I am trying to understand half wave symmetry. I understand the first (a) graphical image is half wave symmetry but (b) seems ...
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1answer
21 views

How to show: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$

I am struggling to show below in a big question: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$ Tried to analyse with geometric ...
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2answers
47 views

Heat flow equation via Fourier Series

I know how to solve heat equations and wave equations defined on $\mathbb{R}^n\times(0,\infty)$ using Fourier transform. But I am having trouble solving similar equations defined on finite intervals ...
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0answers
32 views

PDE Separation of Variables

I'm trying to find a solution to: $v_t = kv_{x x} , 0 < x < l, 0 < t < ∞$ $v(0, t) = 0$ $v_x(l, t) = 0$ $v(x, 0) = −U$ I have: $v(x,t) = X(x)T(t)$ $X(x)T'(t) - kT(t)X''(x) = 0$ ...
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1answer
36 views

$\partial^2_t u(x,t)=\partial^2_x u(x,t)$ - periodic BC

Hi I am looking for a complete solution to the pde given below, it is a hyperbolic pde and I specify the initial conditions and boundary conditions (periodic). Thanks for your help. I show what I do ...
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1answer
31 views

Converting certain complex exponentials to trigonometric functions

The original question is: Show that $$f(x)=\sum_{k=-\infty}^\infty c_k e^{-i kx}=\frac{2\sinh(\pi)}{\pi} \sum_{k=1}^\infty \frac{(-1)^{k-1}k}{1+k^2}\sin(kx)$$ where $\displaystyle ...
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0answers
15 views

Strange variable in discrete Fourier transform definition

Following Wikipedia, we have the next definition of discrete Fourier transform: $ X_k = \sum _{n=0} ^{N-1} x_n e^{{-2 \pi i n k}/{N}}$, where $k$ is an integer ranging from $0$ to $N-1$. Everything ...
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1answer
90 views

Evaluate the series $\sum_{n=1}^{\infty}\frac{\sin(\frac{\pi a}{a+b}n)}{n^3}+\frac{\sin(\frac{\pi b}{a+b}n)}{n^3}$

I have to evaluate the series: $$\sum_{n=1}^{\infty}\frac{\sin(\frac{\pi a}{a+b}n)}{n^3}+\frac{\sin(\frac{\pi b}{a+b}n)}{n^3}$$ Where $a$ and $b$ are real numbers. Since I'm not very good with ...
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1answer
34 views

PDE Using Fourier Series

I'm trying to find the solution to(I don't need to find the coefficient): $v_t = kv_{x x} , 0 < x < l, 0 < t < ∞$ $v(0, t) = 0$ $v_x(l, t) = 0$ $v(x, 0) = −U$ Where U is a constant ...
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2answers
30 views

Fourier series time shift proof?

Prove that if $f(x) \sim \sum c_k e^{ikx}$, then $f(x+t) \sim \sum c_k e^{ikt} e^{ikx}$. Replacing the instance of $x$ with $x + t$, we have that $$f(x + t) \sim \sum c_k e^{ik(x+t)} = \sum c_k ...
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0answers
33 views

When does this fourier series converge?

For which $-2\pi < x < 2\pi$ does this series converge? $$1 = \sum_n^{\infty} A_n\cos\left[\left(\frac{1}{2}+n\right)x\right]$$ The cosine function is piecewise orthogonal. I found $$A_n ...
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0answers
28 views

Convergence of the Fourier series of a continuously differentiable function

I'm taking an introductory course in Fourier analysis and I'm trying to solve the following problem Prove that the Fourier series of a continuously differentiable function $f$ on the circle is ...
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1answer
40 views

Fourier cosine series for $\cos x$

I was trying to find the Fourier cosine series of the function $\cos x$ in $[0, \pi]$. But I am getting all $a_n$ zero. How to proceed?
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1answer
21 views

Finding the Fourier series coefficients of $ \sin(4 \pi t) $

I'm trying to find the Fourier coefficients of $ \sin(4 \pi t) $ I thought I knew how to do it, working backwards with Euler's formula, but when I check my answer I'm off by a negative. I said that ...
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0answers
38 views

Derive the characteristic function of the standard normal distribution N(0,1)

A: Derive the characteristic function $\phi (u)$ of the standard normal distribution N(0,1) by solving: $\int_R e^{iux} f(x) dx$ where $f(x)$ is the probability density function of $N(0,1)$ and ...
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1answer
15 views

Show that there is $C_1,C_2$ s.t. $C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$ where $D_N$ is the dirichelet kernel.

Let $D_N=\frac{\sin(\pi(2N+1)x)}{\sin(\pi x)}$ the dirichlet kernel. Show that there is $C_1,C_2$ s.t. $$C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$$ where $\mathbb S^1=\mathbb R/\mathbb ...
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0answers
17 views

Intuition behind orthogonal sin functions

When deriving Fourier series, an important step is to establish that the integral of the product of two periodic sin functions is 0 if they have a different frequency. This then allows you to define ...
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1answer
13 views

Algebra to Series Simplification

From the Wikipedia page on Fourier Series         $\begin{align} \frac{1}{\pi}\int_{-\pi}^{\pi}\frac{\pi}{x}\sin(nx)dx & = -\frac{2}{n \pi}\cos(n \pi) + \frac{2}{n^2 \pi^2}\sin(n \pi) \\ & = ...
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0answers
14 views

Piecewise nonhomogeneous PDE

The problem is $$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$ $$0<x<\pi \\ t>0$$ $$f(x)=\begin{cases} 50 & 0<x<\frac{\pi}{2} \\ 0 & ...
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0answers
22 views

What are the conditions (like Dirichlet conditions) of Fourier series of complex valued function

Are there any conditions to expand a complex valued function to Fourier series? For example, $w(z)$. $z$ is complex variable. $w(z)=\sum_{k=-\infty}^\infty \phi_k(z)w_k $ , where $\phi_k $ are ...
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2answers
26 views

Finding the zero-state output

The input and output of a stable network are related via the following equation. $$\frac{d^2y(t)}{d(t)} + \frac{2*dy(t)}{d(t)} + 10y(t) = \frac{dx(t)}{d(t)} + x(t)$$ x(t) = input, y(t) = output, ...
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0answers
24 views

Change of variable in Fouriers series

I am trying to reconcile two definitions of the Fourier-cosine series that I have. The first definition is \begin{equation} f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos \Big( ...
2
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1answer
38 views

Differentiability of Fourier Series

The following Fourier Series is continuous (by M-test) $$ \sum_{n=1}^{\infty}{\frac{\sin{n^n x}}{n^n}}$$ Is it differentiable anywhere, and if so where? http://kryakin.org/at/hardy_1916_W.pdf covers ...
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1answer
29 views

How to calculate the Fourier series of $\sin x-4\sin 3x+7$

How to calculate the Fourier series of $\sin x-4\sin 3x+7$ I obtain $0$ por the an and bn coeficients, and I think that's incorrect...
2
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1answer
65 views

Rudin's Real & Complex - Q9.11 (Fourier)

I have solved most of Question 9.11 of Big Rudin : Find conditions on $f$ and/or $\widehat{f}$ which ensure the correctness of the following formal argument : If $\varphi(t) ~=~ ...
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0answers
20 views

Prove that every periodic function can be written as a Fourier series

I was wondering whether there is a proof that every periodic (period length $\ell$) function can be written as a Fourier series? $f(x) = a_0 + \sum_{\nu=0}^\infty \lbrace a_\nu * cos(\nu \frac{2 ...
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1answer
91 views

Evaluate the sum $\sum_{k=1}^\infty \big(\frac{1}{36k^2-1}+\frac{2}{(36k^2-1)^2}\big)$

I have to evaluate the series: $$\sum_{k=1}^\infty \left(\frac{1}{36k^2-1}+\frac{2}{(36k^2-1)^2}\right).$$ I tried using the identity $\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$ but I got ...
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0answers
15 views

Show that $ \sum_{\mathbb{Z}/N\mathbb{Z}} f(n) g(n+r)h(n+2r) = \sum_{a \in \mathbb{Z}/N\mathbb{Z}} \hat{f}(a)\hat{g}(-2a)\hat{h}(a)$

I found this Fourier series identity in a book on Harmonic analysis but the proof is inclear. Maybe it makes more sense using bra-ket formalism. $$ \sum_{r, n \in \mathbb{Z}/N\mathbb{Z}} f(n) ...
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0answers
36 views

Fourier Series expansion condition

I know that to expand any function by Fourier series it has to agree a set of conditions called Dirchlet Condition. My question is why cosec(x) cannot be expanded. Which of the conditions dont agree. ...
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2answers
38 views

Fourier Tranform of piece wise function

I want to find the fourier transform of this input signal. Let the unit function unit (t, a, b) have the value 1 on the interval a≤ t < b and the value 0 otherwise. f(t) = (t)unit(t, 0, 0.5) + ...
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1answer
20 views

Why does a function need to be bounded in order to have a Fourier series?

Suppose I have some piecewise smooth periodic function $f$. Why does $f$ need to be bounded in order to have a Fourier series representation? Couldn't we consider the interval that it's unbounded as ...
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0answers
23 views

Parseval's identity on the a half real line

I have two question related to the Fourier transform and Parseval's identity. If $f$ is an integrable function in $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ and $f$ has compact support, doest its ...
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0answers
32 views

Linear independent set of function applied to water waves.

I need to show that a given surface elevation $\zeta(x,y,t)$ defined on a closed region $D(x,y,t): 0<x<L_x,0<y<L_y,0<t<T$ and not periodic on D: $$ ζ(x,y,t) = \sum_{n=1}^{\infty} ...
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1answer
63 views

Sum of complex exponential [closed]

Is the following sequence of partial sums bounded? $$\sum_{n=1}^{N}{e^{i\,n!\,x}}$$ where $x$ is in $\left(0,2\pi\right)$ and $x$ is not a rational multiple of $\pi$.
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26 views

Half-range Fourier Sine Series of $4 - x^2$

I need your help in checking my working for finding the half-range Fourier Sine Series of $$f(x) = 4 - x^2, 0 \le x \le 2$$ Now, since this is looking specifically for the half-range Sine series, I ...
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0answers
34 views

Fourier Series Representation of $t + 1$

I need your help double-checking my working for finding the Fourier Series representation of a "piecewise" function. The function I was given is $$f(t) = \begin{cases}t + 1 & ,-1 < t \le 1 \\ ...
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1answer
41 views

In which sense is the Fourier transform the continuous analogue of the discrete Fourier transform? A mathematical point of view.

I try to understand in which sense the Fourier transform is the continuous analogue of the discrete Fourier transform. I know, there are many books and many questions on this site concerning this ...
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0answers
17 views

Parseval's theorem derivation using Bessel's inequality

Could someone reiterate how to derive Parseval's theorem from Bessel's inequality? I'm just a bit confused from the text in the textbook and would appreciate some clarification! Thanks in advance ...
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1answer
29 views

How Does This Fourier Grapher Work?

A friend sent me this link: http://toxicdump.org/stuff/FourierToy.swf. I am not very versed in fourier series. I know the basic definitions and some convergence stuff, what you'd learn in a basic ...
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0answers
24 views

Pointwise convergence of a Legendre polynomial expansion

$\langle\cdot,\cdot\rangle$ is the dot product on the real vector space $\mathcal C ([0,1],\mathbb R)$ defined by $\langle f,g\rangle = \int_{-1}^1 fg$, and $(L_n)$ is the family of normalised ...
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1answer
68 views

Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.

Theorem 2.1 : Suppose that $f$ is an integrable function on the circle with $\hat f(n)=0$ for all $n \in \Bbb Z$. Then $f(\theta_0)=0$ whenever $f$ is continuous at the point $\theta_0$. Proof ...
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0answers
15 views

Is it possible to simplify this function I. Series form?

I have to simplify this function, $w(x,y)=\sum\limits_{n=1}^{\infty} \sum\limits_{m=1}^{\infty}\dfrac{16q_0}{(2m-1)(2n-1)\pi^6D}\Big[\dfrac{(2m-1)^2}{a^2}+ \dfrac{(2n-1)^2}{b^2} \Big]^{-2}\cdot ...
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2answers
24 views

How do you know if a function can be represented as a convergent power series in terms of analyticity and singularities?

I was reading my textbook on Fourier Analysis and it reads, "Let $F(z)$ be an analytic function of the complex variable $z = x + iy$, without singularities for $|z| \leq 1$. Then $F(z)$ can be ...
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1answer
43 views

Calculating Fourier coefficients [closed]

I am unable to get $2^{-|k|}$ as the Fourier coefficients of $\frac {3}{5-4\cos(x)}$ on $[0,2\pi]$ Kindly give me some clue as how to get this value $2^{-|k|}$. i am using the formula to find ...
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0answers
27 views

Where is this formula In the Fourier series coming from?

I can't understand where this $x\frac{pi}{L}$ is coming from in the Fourier series for any function with period $2L$, any help would be great.
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1answer
11 views

DFT trigonometric interpolation of $\log(x+1)$

I am hoping that someone could explain to be a concept of trigonometric interpolation and why the it looks as follows for $\ln(x+1)$. I am not sure of the understanding as to why there is such ...
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0answers
18 views

Fourier Transform of triangle function 𝑥(𝑡)=Δ((t-1)/2)

Can you please tell me if my working is right for the fourier transform of this function: 𝑥(𝑡)=Δ((t-1)/2) My workings are: my workings I have used the fourier transform standard results. Please ...
2
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0answers
61 views

if $f(x)$ is periodic $\left|\int_1^\infty f(x) x^{-s} dx\right| \sim C\left|\int_1^\infty \sin(x) x^{-s} dx\right|$ when $\text{Im}(s) \to \infty$

is it true that if $f(x)$ is periodic, non-constant and bounded $$\text{when } T \to \infty ,\qquad\qquad\sup_{|t| \ <\ T}\ \ \left|\ \int_1^\infty f(x) x^{-\sigma-it} dx\ \right| \ \sim \ ...
1
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1answer
49 views

trigonmetric integral (might be related to complex variable or Fourier series)

Let $p\in \mathbb{N}.$ I would like to know how to compute $$\int_{0}^{2\pi} \cos^j(t) \cos(kt) dt, \quad \quad j,k\in \{1,2,\ldots,p\}.$$ Could somebody help me with this? It might be related to ...
1
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1answer
51 views

Hard Integral from a special case of Fourier series

I was trying to find the equation of a sinusoidal function that has one upward facing semicircle and immediately after one downwards facing semicircle which would make up a period of the function. ...