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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Computing the complex fourier series for triangular wave from trigonometric coefficients

I'm trying to figure out how to compute the complex Fourier series for the triangular wave, given the trigonometric coefficients. The book gives as a result for the complex series the following: $$ ...
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14 views

Heat and Wave equation - Green's function versus Fourier series?

I am learning how to solve the heat and wave equation in bounded domains in 1 and 2D as well as in $\mathbb{R}$ and $\mathbb{R}^2$. In the latter case I have learned the representation formulas i.e. ...
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2answers
100 views

Why does specifying an interval for a function make the function odd or even?

I am currently reading about Fourier series and Orthogonality of functions and Complete Sets of functions. Below are two extracts from the book I'm reading for which I simply do not understand: ...
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1answer
23 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) ...
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1answer
58 views

Show that $\sup_{x \in \mathbb{R}}|\sigma_nf(x) - f(x)| \leq C\frac{\log n}{n}$, for $f$ $2\pi$-periodic and Lipschitz.

I'm learning about Fourier analysis and need help with the following 2 problems: $(1)$ Show that $\forall t \in (0, \pi], K_n(t) \leq \min\{n +1, \frac{\pi^2}{(n + 1)t^2}\}$. $(2)$ For the ...
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1answer
20 views

Expanding a function in a Fourier Series

I am having an issue integrating the sin function with the variable of n, any help would be appreciated. I have deduced it to an odd sine series with the following for B_n and I am unsure how to ...
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3answers
92 views

Solving Viscous Burgers using spectral method

I am trying to solve the Viscous Burgers equation using the spectral method. $$u_t+uu_x = Du_{xx}$$ where $D$ is a constant (chosen to be zero) and with the initial condition $$u(x,0) = ...
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1answer
22 views

Fourier coefficients of a triangle function

I'm trying to find the Fourier coefficients ($c_n$), of the following function : for $x$ in $[-\pi/2;\pi/2[$ $f(x)=x$ for $x$ in $[\pi/2;3\pi/2[$ $f(x)=\pi-x$ I think its not that hard, but I keep ...
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36 views

Classical Full Fourier Series of f(x) converges uniformly to f(x)

Prove the classical full Fourier series of $f(x)$ converges uniformly to $f(x)$ if $f(x)$ is continuous of period $2\pi$ and its derivative $f'(x)$ is piecewise continuous. How do I go about doing ...
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1answer
34 views

If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

I'm learning about Fourier analysis and need help with the following problem: Suppose $f \in L^1[-\pi, \pi]$ and $\alpha_n, \beta_n$ are the Fourier coefficients of $f$. Show that if $f$ is odd ...
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16 views

Fourier transform for spectral method equation explanation.

Consider the advection equation: $$u_t+u_x =0$$ Using the spectral method compute the Fourier transform of $U(x_j,t)$ which will give us an approximation for the spacial derivative. The Fourier ...
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1answer
14 views

Laplace equation with boundary conditions in polar coordinates

Show that the problem with this boundary conditions $u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}=0$, $\quad 0 < r < 1, \quad 0 < \theta < \pi$ $u(r,0)=0$ $u(r,\pi) =T_0$ $u(1,\theta) =T_0 $ ...
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1answer
19 views

Quick Fourier Series help?

I was given a graph (shown above) and was asked to represent this as a Fourier Series. I was able to solve $a_0$ with no problem. However, when I was integrating for $a_n$ and $b_n$, I was having a ...
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1answer
51 views

Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots = \frac{1}{2}\cot\frac{x}{2}$ for $x \neq 2k\pi, k \in \mathbb{Z}$

I'm learning about Fourier series (specifically Cesàro summation) and need help with the following problem: Show that the Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots$ is equal to ...
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20 views

Laplace equation with Boundary value conditions by parts

I don´t know how to procced in this problem by parts $u_{xx}+u_{yy}=0$, $\quad 0 < x < \pi, \quad 0 < y < \pi$ $u(x,0)=0$ $u(0,y) = \begin{cases} y, & \text{for } 0 < y < ...
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1answer
19 views

Fourier Series - Convert sinusoidal form to exponential form

How do I get from $-2j=\cos(\frac{\pi}{2})+2j\sin\frac{\pi}{2}$ to $-2j=2e^{-j\frac{\pi}{2}}$ Background: Given $x(t)=10+3\cos\omega_0t+5\cos(2\omega_0t+30°)+4\sin3\omega_0t$, its period is ...
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1answer
22 views

Fourier Series of $f(x)=1$, can I do it for $(-\infty, \infty) $?

I am trying to approximate the line $y=1$ by fourier series. I can see a lot of examples where we define the domain for $x$. However, Is it possible to define the series everywhere? For example if ...
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1answer
38 views

Mean-Square Fourier Convergence

Let $ \left \{X_n\right \} ^{\infty}_{n=1}$ be any orthogonal (in the $L^2$ sense) set of functions. Let $$S_N(f) = \sum^{N}_{n=1} \frac{(f, X_n)}{ \left \|X_n\right \|^2} X_n$$ be the “Fourier ...
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1answer
28 views

How to solve this Laplace boundary value problem by Fourier series

can someone help me?, I don't know how to proceed in the last boundary condition $u_{y}(x,1)=x(1-x)\ $ $u_{xx}+u_{yy}=0\ $, $\ 0<x<1,\ 0<y<1$ $u(0,y)=0$ $u(1,y)=0$ $u_{y}(x,0)=0$ ...
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1answer
28 views

Using Fourier Transform to solve an ODE

Consider the differential equation $$f^{iv}+3f^{''}-f=g$$ I have read that taking the Fourier Transform of both sides gives ...
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22 views

$f(t)=\sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}$ for $f\in L^{2}(\mathbb T)$?

Let $f\in L^{2}(\mathbb T).$ Define $g(t):= \sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}, (t\in \mathbb T).$ Since $\hat{f} \in \ell^{2}(\mathbb Z),$ we note that $g\in L^{2}(\mathbb T).$ My ...
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13 views

Understanding the spectral method

It is now clear how to solve a single linear, constant coefficient Q-th order PDE,say $$U_t + P(∂/∂x)U = 0, U(x, 0) = f(x)$$ where P is a polynomial of degree Q. The ODEs for the Fourier coefficients ...
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11 views

Why this pulse Fourier Series MATLAB code is wrong?

I try to create a MATLAB script that plots a Fourier Series of a pulse of width d and period T. ...
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14 views

Decay of fourier series implies existence of (non contiuous) derivative

Let $a_n=\mathcal{O}(\lvert n\rvert^{-\alpha})$ where $\alpha>\frac{3}{2}$ then \begin{equation} f(x):=\sum_{n\in\mathbb{Z}}a_n e^{inx}, \end{equation} is differentiable and its derivative is in ...
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1answer
36 views

Finding the energy of the nth harmonic $h = h_n$ for the wave equation

Q: A string of tension $T$, density $\rho$ with fixed ends at $x = 0$ and $x = \ell$ is hit by a hammer so that $u(x,0) = 0$, $u_t (x,0) = V$ in $[-\delta + \frac{1}{2}\ell, \delta + ...
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4 views

Can a Fourier series have a recurring “sub-sequence” in a single period?

Let $f(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \sin(\frac{2\pi}{P}nx + \phi_n)$ be a Fourier series with period $P$. Obviously $f(x)$ is repeating every period $P$, but I'm wondering if within one full ...
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26 views

Using Fourier Series to find formulas for f(x)

Given $$ f(x) = \begin{cases}x+1,&-1<x<0\\x,& 0<x<1\end{cases} $$ and $$ f(x+2)=f(x), $$ I am asked to find the formula for $f(x)$ in the intervals $1<x<2$ and ...
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15 views

check if two functions $f(x)$ and $g(x)$ are equal in an interval $[a,b]$

Is it possible to check if two functions $f(x)$ and $g(x)$ are equal in some interval $[a,b]$, that is to check if $f(x) = g(x)$ for $a \leq x \leq b$? My first idea was to check if $\int_a^b (f(x) - ...
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2answers
41 views

Find Fourier Transform without the use of integration?

If I have $\mathcal{F}(f(t))=F(\omega)$ with $$f(t)= \begin{cases} 1, & 0 \leq t < \pi\\ -1,&- \pi \leq t <0 \\ 0 & \text{otherwise}\end{cases}$$ I have found $F(\omega)= ...
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1answer
37 views

Finding the Fourier series of an absolute value function.

I have to find the Fourier cosine series of $ | \sin x |$ on $(-\pi, \pi)$. I understand the trick to use is that the integral of an absolute value function is even. The formula for the coefficients ...
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2answers
111 views

Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
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1answer
38 views

Error in the statement of Wirtinger's inequality?

Theorem. Suppose that $f(x)$ has a continuous derivative on the interval $[0, 1]$, and that $\int_0^1 f(x)\, dx=0$. Then $$\int_0^1 |f'(x)|^2\, dx\ge 4\pi^2\int_0^1 |f(x)|^2\, dx.$$ Proof. ...
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1answer
56 views

Fourier series for $f(x)=\sin(ax)$ where $a$ is not an integer?

I was wondering if anyone could help me with this fourier series problem? Expand the following function in Fourier cosine series: $f(x) = \sin(ax)$ $(0\le x \le \pi)$ , where $a$ is not an ...
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1answer
36 views

Proof that $\|S_N\|_p < \infty $ is equivalent to $\|S_N f - f\|_p \to 0$ as $N \to \infty$

I am having difficulties with the proof of proposition 1.9 in the book "Classical and multilinear harmonic analysis, Vol. 1" by C. Muscalu and W. Schlag. The following statements are equivalent ...
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2answers
123 views

Show that $\frac{\pi^2}{12} = \sum^\infty_{k=1}\frac 1 {k^2}$ using Fourier series

So i Have created a Fourier as $$f(x)=\frac{1}{3} + \sum^{\infty}_{n=1}(\frac{-4}{n^{2} \pi^{2}}\cos(n \pi x))$$ and i believe i can rearrange this to: $$ f(x) = \frac{1}{3} - ...
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0answers
26 views

Help solving a Fourier Series for a function with P=2L

I have a periodic function $$ f(x) = \begin{cases} -2x-x^2, & -2 \leq x < 0 \\ 2x-x^2, & 0 \leq x < 2 \\ f(x)=f(x+4) & otherwise \end{cases}$$ and this period is repeated. I ...
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1answer
28 views

Using Fourier Series to find the sum of a numerical series

I have to use a Fourier series to compute the sum of the series $$\frac{1}{2} + \sum_{n=1}^\infty (-1)^n \frac{1}{n^2 + a^2}$$ My guesses are the Fourier series $$e^{ax} = \frac{e^{a\pi} - ...
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1answer
17 views

Prove the following about the integral of Fourier coefficients

I'm having a difficult time going from $$\sum_{n=1}^\infty (\cos nx \int_{-\pi}^\pi f(t) \cos nt dt + sin nx \int_{-\pi}^\pi f(t) \sin nt dt)$$ to $$\sum_{n=1}^{\infty}(\int_{-\pi}^\pi f(t) \cos ...
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17 views

Application of Abel's Method to Summation of Fourier Series Question

The series $$f(x, r) = \frac{a_0}{2} + \sum_{n=1}^\infty r^n (a_n \cos nx + b_n \sin nx)$$ where $0 \le r \lt 1$ clearly converges, as the terms monotonically decrease. My question: My textbook ...
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3answers
70 views

Evaluate this integral

Evaluate this integral: $$a_{n} = \dfrac{1}{\pi}\int^{\pi}_{-\pi} \left(\dfrac{T}{2\pi}\right)^2y^2\cos ny ~dy$$ I understand this needs to be integrated by parts, so far I have $a_{n} = ...
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1answer
36 views

Use integration by parts to verify the following :

Using integration by parts show that: $\int^{1}_{-1}P_{n}\left( x\right) P_{m}\left( x\right) dx$ = $\dfrac {2}{2n+1}, m=n$ and $0$ if $m\neq n$ Where the functions are both Legendre polynomials. ...
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2answers
40 views

Calculating the Fourier Series of a discontinuous function

Let $f\left(x\right)=\begin{cases} 1,& \text{if } 0<x<\pi\\ 0, & \text{if } \pi<x<2\pi \end{cases}$ $f\left(x+2\pi\right)=f\left(x\right)$ I have worked out ...
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1answer
18 views

Obtain Fourier Coefficients from Discrete Fourier Transform

The Fast Fourier Transform $y[k]$ of length $N$ of the length-$N$ sequence $x[n]$ is defined as: $$y[k] = \sum_{n=0}^{N-1}e^{-2\pi i \frac{k n}{N}}x[n].$$ I want to know how are the $y[k]$ related ...
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1answer
28 views

Limit of a sequence of functions recursively defined by integrals

$f_n:[0,\infty)\to\mathbb{R}$ is defined recursively by $f_1:=0$ and $$f_{n+1}(x)=e^{-2x}+\int_0^xf_n(t)e^{-2t}dt,\qquad n\ge 1$$ I need to show that the limit $f(x):=\lim_{n\to\infty} f_n(x)$ exists ...
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29 views

Fourier series converges “almost everywhere”

I'm reading "Fourier series" by Rajendra Bhatia. At one point, the author says: "[..]one can show the existence of a continuous function whose Fourier series diverges except on a set of points of the ...
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3answers
67 views

Fourier analysis — Proving an equality given $f, g \in L^1[0, 2\pi]$ and $g$ bounded

We were given a challenge by our Real Analysis professor and I've been stuck on it for a while now. Here's the problem: Consider the $2\pi$-periodic functions $f, g \in L^1[0, 2\pi]$. If $g$ is ...
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1answer
65 views

Find the Fourier series of the trigonometric polynomial $f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx})$

I'm learning about Fourier series and need help with this problem: Given the trigonometric polynomial $$ f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx}) $$ find the Fourier ...
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1answer
31 views

Multidimensional Fourier-cosine series

The 2D fourier-cosine series on $(-\pi,\pi)\times(-\pi,\pi)$ is given by \begin{equation*} f(x_1,x_2) = \sum_{n_1=0}^{\infty} \sum_{n_2=0}^{\infty} a_{n_1,n_2} \cos(n_1x_1)\cos(n_2x_2) ...
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2answers
36 views

Compare $L_2$ norm of Fourier series of $ f$ and $ f'$

Let $f$ be a periodic continuous function on $[0,2\pi]$ with $f'$ continuous. Let the Fourier series of $f$ be ${a_o\over 2} + \sum_{n=0}^\infty(a_n\cos nx + b_n \sin nx).$ Is it true that ...
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43 views

Verifying work on Fourier series

I'm learning about Fourier series and need some help with this following problem: Consider the function $f(x) = \frac{\pi - x}{2}, \ x \in [0, 2\pi)$ extended periodically with period $2\pi$. Find ...