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

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12 views

Vanishing fourier coefficients

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. I want to show that $$\widehat{g}(l)=0 \ \text{for} \ l \not\equiv 0 \ \text{mod} \ n,$$ where $$ \widehat{g}(l) = ...
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1answer
10 views

Complex Fourier Series coefficient reduction.

I am trying to understand the Complex Fourier series solution for the following function, as printed on "Fundamentals of Electric Circuits" by Alexander & Sadiku: The solution printed on the ...
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2answers
45 views

$\int_{-L}^{+L}h(z)\,dz = 2 \sum_{-\infty}^{+\infty}\frac {a_n}{n} \sin (nL)$

is it possible to find a formula for $a_n$ from $$\int_{-L}^{+L}h(z)\,dz = 2 \sum_{-\infty}^{+\infty}\frac {a_n}{n} \sin (nL)$$ For $n=0$ the series is $0$ Thanks
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0answers
31 views

Linear ODE and Fourier Series

Let $m,k_0,k$ be positive real numbers and $x_1$, $x_2$ be real-valued functions of time. Suppose we have following system of two coupled ODEs ( motivated by a coupled oscillator with two masses ...
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0answers
21 views

Fourier series - Understanding an equality

Why is this equality true: $$\left\langle {f,g} \right\rangle = \sum\limits_{n = - N}^N {\hat{f}(n)\hat{g}(n)}$$ where $$f = \sum_{n=-N}^N c_n e^{int}, g=\sum_{n=-N}^N d_n e^{int} $$ and ...
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3answers
103 views

Compute the fourier coefficients, and series for $\log(\sin(x))$

I posted a similar question with a bad response, so I am retrying with hopes of better knowledge. The fourier series is in the form: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + ...
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0answers
48 views

Fourier series for logarithm of sine.

I looked up here: Fourier series of Log sine and Log cos I have modified the question: How can I derive the coefficient $a_n, b_n$ for $\log(\sin(x))$ in the fourier series representation? Also, I ...
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1answer
28 views

Schwartz space on $\mathbb T^{n}$

For the definition of Schwartz space space on $\mathbb R^{n},$ see this. My Questions: (1)Is it make sense to talk of Schwartz space on torus $\mathbb T^{n}$ ? If yes, what can be the analogous ...
2
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0answers
45 views

A deep understanding of the Fourier transform

I feel like i don't understand the Fourier transform. I've seen what it does and its properties but even after reviewing various proofs i don't understand why we end up explicitly with a relation ...
2
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0answers
79 views

how to use Matlab ifft to calculate the following integral? [duplicate]

$$R(t)=\int_{-\infty}^\infty\dfrac{\omega e^{i\omega t}}{(3-\omega^2)^{2}+4\omega^2}\,d\omega$$ where t is a integer and $t>0$ I used to calculate this integral by numerical integral,but it seems ...
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1answer
8 views

Conversion of Fourier Basis

I find myself working with a time series of length n composed of sparse frequency data. I would like to extend this to a time series of length m > n that has the same sparse frequencies at the same ...
3
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1answer
42 views

A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
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0answers
17 views

Plancherel's theorem variants

How would you prove a variant form of Plancherel theorem: If $(c_n)_{n\in\mathbb{Z}}$ are coefficients and $\sum_{n\in\mathbb{Z}}|c_n|^2<\infty$, then there exists a unique function $g\in L^2(0,1)$ ...
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0answers
31 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
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0answers
14 views

Solution of a differential equation with problem of Cauchy

The question is the next: What can I say from the existence, uniqueness and continuos dependence of the solution? Is this a strongly continuos one-parameter group or a semigroup. $ \left\{ ...
0
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1answer
36 views

2D linear inhomogeneous wave equation with inhomogeneous time-independent initial conditions

I'm looking for any insight into solving the following PDE: $$u_{tt}=c^2 (u_{xx}+u_{yy})-\sin(y)$$ $$u=0, y\in {0,\pi} $$ $$u_x=0, x\in {0,1}$$ $$u(x,y,0)=\cos(\pi x)\sin(3y) $$ $$u_t(x,y,0)=0$$ ...
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0answers
23 views

Function of bounded Fourier degree; bounding on subinterval

Suppose $f(x)=\sum_{|k|\le d} a_ke^{2\pi i kx}$, and it is given that $|f(x)|\le 1$ on $[0,L]$. Over all such functions, what is the maximum possible value of $$\max_{x\in [0,1]} |f(x)|?$$ (For ...
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1answer
50 views

Fourier series of this aperiodic piecewise function

I am trying to get Fourier sine series for $$ f(x) = \left\{ \begin{array}{lr} 3 & : 0\le x\le 6\\ 3-x & : 6\le x \le 9 \end{array} \right. $$ So far I know that the ...
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0answers
10 views

Laplace Equation- Dirichlet Problem- superposition approach?

I have tried to treat this as a seperable partial differential equation but I can't seem to get an equation for the product functions.
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0answers
13 views

Fourier series qn determine the fourier series coefficients

Can someone please help me with this Fourier series $q_n$: determine the fourier series coefficients of $x(t)$ given as $x(t) = \cos4t + \sin8t+3$?
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0answers
21 views

given a solution of a second order ODE, what is the way to find another linearly independent solution?

So if i'm given a second order linear diff eq. and one of its solutions, what is the way to find another linearly independent solution? Thanks in advance!
3
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2answers
53 views

Examples of orthonormal bases for $L^2[0,1]$ that are not trigonometric?

What are examples of orthonormal bases for $L^2([0,1],dx)$? For instance, the following trigonometric polynomials are orthonormal basis $$\left\{1, \sqrt{2}\sin(2\pi jx),\sqrt{2}\cos(2\pi j ...
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1answer
35 views

Verifying Distribution Equivalence for Fourier Series Expansion

In my lecture notes, given a periodic distribution $T \in (C_{per}^\infty([-\pi,\pi]^n))'$, the Fourier coefficients are defined by $$\hat T(m) = T({1 \over (2\pi)^n}e^{-i m \cdot x}),$$ for $m \in ...
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27 views

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why?

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why ?
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0answers
29 views

Find the Fourier Series of the function?

Would someone be able to help me solve this? The function $f:(0,\pi]->\mathbb{R}$ is defined by; $$f(x) = \begin{cases} x & 0 < x \le \frac\pi2 \\[5pt] 0 & \frac\pi2 < x \le \pi ...
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1answer
24 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period ...
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1answer
10 views

Fourier series Even vs. Odd and effect of integral bounds?

I understand that when you express a function in fourier series there are 3 coefficients you need to calculate ( a0, an, bn) and I have in the past made use of the symmetry of the function in my ...
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1answer
16 views

How to use trigonometric Fourier series to verify this result

I'm studying signal processing. I've found the associated Fourier Series for a message $m(t)$ = $t^2$ over the interval $[-1, 1]$ with period $T = 2$. However, I'm then asked to verify that ...
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2answers
40 views

Integrating a Fourier series

I am trying to integrate the Fourier series of $$f(x) = x,-\pi<x<\pi.$$ Using complex exponentials to find the series, I get the series $$\frac{2}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} ...
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1answer
32 views

Odd or Even for Fourier Series?

I have the function $f(x) = -x^2 + x\pi$ and $0\le x\le \pi$ and without seeing the graph I want to show if it is odd or even, but of course $f(x) = f(-x)$ doesn't show that it is even because I can't ...
2
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1answer
79 views

Evaluating $\int _{-\pi}^{\pi}x^2cos(nx)dx $

Hello I'm trying to evaluate $$\int_{-\pi}^{\pi} x^2\cos(nx)dx$$ I understand you have to apply integration by parts twice but I always get zero and I know this is wrong. I always end up with ...
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1answer
42 views

Find the Fourier transform of $\frac1{1+t^2}$

Find the Fourier transform of $$f(t)=\frac1{1+t^2}$$ using contour integration that $$F\{f(t)\}=\int^\infty_{-\infty}\frac1{1+t^2}e^{2\pi ft}dt$$ How can I do this?
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16 views

Fourier series for asymmetric intervals

I have to compute fourier series for functions like I know how to compute fourier series for functions defined on intervals $[-L,L]$ but this function is defined in an "asymmetric" way. I've ...
2
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1answer
58 views

Prove a trigonometric series is positive

Let $f(x)= \sum_{n=-\infty}^\infty \frac {e^{inx}}{1+n^2}$ on $[-\pi,\pi]$. Prove $f(x)>0$ for $x\in[-\pi,\pi]$. This is an review question for my Fourier course. I am not sure how to approach ...
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0answers
23 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, ...
2
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2answers
71 views

Why am I allowed to set a fixed point in a fourier series?

I'm working with $f(t)=\cos(at)$, for $a\in (0,1)$, on the interval $(-\pi,\pi)$. I've calculated the fourier series on this interval. what I would want to do next is to fix $t=\pi$ and get a nice ...
3
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1answer
75 views

Relation between Fourier components of a positive function

Here's a problem that has recently come up in my physics research: Let f be a function on [0, 2 $\pi$], which yields positive real numbers. Let the integral of $\int_0^{2\pi}f(x)= 1$. (Just for the ...
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1answer
66 views

Assume that $f$ is $2\pi$ continuous and $C^1$ such that $\int_{-\pi}^{\pi} f(x) dx=0$.

Show that $\int_{-\pi}^{\pi} (f(x))^2 dx \leq \int_{-\pi}^{\pi} (f'(x))^2 dx$. So here's my approach to this question: Assume that $f$ was $2\pi$ continuous and $C^1$. Therefore, we have that ...
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0answers
14 views

Fourier series, even and odd n properties

I recently started learning about Fourier series, so I'm still kind of shaky on the topic Given $f(t)=f(t+T)$ and $$f(t)=\sum_{n=-\infty}^{\infty}F_ne^{jn \omega_0t}$$ Show that: (a) If ...
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1answer
18 views

Uniqueness of solution for seperation of variables solvable PDEs

I am taking first course in PDEs and the only way i know of solving PDEs is separation of variables , and all the equations i saw had unique answers due to the ICs and BCs , but not this one : $$ ...
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1answer
29 views

After calculating Fourier series coefficients for $x(t)=2 cos(4t) + 4 sin(10t)$, why am I getting all zeroes for all coefficients?

I am trying to find the Cosine/Sine Fourier series coefficients for the given equation: $$x(t)=2\cos(4t) + 4\sin(10t)$$ $\cos(4t)$ has a period of $T=\frac{\pi}{2}$, and $\sin(10t)$ has a period of ...
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1answer
24 views

Integral equality $\int_{-\pi}^\pi\dots = \int_{|t|\le \delta}\dots+\int_{\delta\le |t|\le \pi}\dots$

This is an excerpt from here (page 6, bottom) I don't know if this is a typo or not, but what exactly happened to the integral of $\int_{-\pi}^{-\delta}$ for the $|\sigma_n(x) - f(x)|$? I don't ...
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1answer
48 views

Prove $\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}$ with Fourier series

I want to prove $$\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}\;x\in(0,2\pi)\;\;\;\;[1]$$ I have two questions regarding this: $(1)$ How can I find a function $f$ such that the former ...
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0answers
4 views

Short-Time-Fourier-Transform: why overlapping the window?

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I ...
3
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1answer
41 views

Fourier series for $f(x)=\begin{cases} 0 & -\pi\leq x<0 \\\sin x & 0\leq x\leq \pi \end{cases}$

Find the Fourier series for $$f(x)=\begin{cases} 0 & -\pi\leq x<0 \\\sin x & 0\leq x\leq \pi \end{cases}$$ I found an answer, I'm not completly sure if it's right. The solution would ...
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0answers
16 views

Fourier transform of $|x|^{-s}$

Using the definition of Fourier transform $\hat{f}(p) = (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(x) e^{ix \cdot p} \ dx$ where $u \in \mathbb{R}^n$. What is the fourier transform of $|x|^{-s}$.
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0answers
22 views

Local behavior of a Fourier series and a intgral

So I have to calculate an integral that involves a Fourier series of some function. I would like to get some kind of local control of the function near zero the series is ...
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1answer
27 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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0answers
12 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
2
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1answer
20 views

Fourier Series Coefficient

I am trying to review the basics. Find the Fourier series for the function $$f(x) =\left\{ \begin{array}{l l} 2x & \quad -\frac{\pi}{2}<x<\frac{\pi}{2}\\ 0 & \quad ...