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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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) $\omega^2=...
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1answer
60 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
96 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 $$u(x,0) = exp(-x/0.2)^2$$...
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23 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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39 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
20 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
20 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
61 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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0answers
21 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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39 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\ $, $\ 0<x<1,\ 0<y<1$ $u(0,y)=0$ $u(1,y)=0$ $u_{y}(x,0)=0$ $u_{...
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1answer
29 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 $$\left(i\lambda\right)^{4}F\left(\lambda\right)+3\left(i\lambda\right)^...
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$f(t)=\sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}$ 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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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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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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15 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 $L^...
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1answer
50 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 $[-\delta + \frac{1}{2}\ell, \delta + \frac{1}{2}\ell]$,...
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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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28 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 $8<x<9$...
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2answers
47 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)= \begin{...
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1answer
46 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
140 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. We ...
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1answer
70 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 integer....
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1answer
38 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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132 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} - \frac{4}{\pi^{2}}\sum^{\...
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0answers
30 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
33 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} - e^{-a\pi}}{\...
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1answer
18 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 n(t-x)...
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20 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
71 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} = \dfrac{T^2}...
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1answer
41 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
43 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
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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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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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72 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
75 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
34 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) \end{...
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2answers
37 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 $$||f-{a_0\...
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0answers
49 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 ...
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0answers
16 views

Gauss Relation. The Discrete Fourier Transform

I am reading a chapter in my book about Discrete Fourier Transformation. I get to this part $$\sum ^{n-1}_{k=0}\xi^{ak}= \begin{cases} & n \text{ if } \xi^{a}=1 \\ & 0 \text{ otherwise} \...
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1answer
40 views

When is solution to Laplace's equation on circle sector $C^2$?

I am working out an example from Pinchover's PDE book (p. 198). Suppose we look for solutions to $\Delta u = 0$ on the circular sector $\{ (r, \theta) \mid 0 < r < a, 0 < \theta < \gamma \...
4
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1answer
57 views

Closed form for $\int^{\pi}_0 \frac{\sin^2 (y)}{a+\cos(y)} \cos(ny) dy$ for integer $n$

I encountered this integral when trying to obtain a Fourier series for the function inside (in connection to this question). Mathematica gives the following general solution (only valid for $|a|>...
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1answer
40 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 ...