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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21 views

Why is $\|f-s_n(f)\|_2=\inf_{T\in\mathcal{T}_n}\|f-T\|_2$

I am working through some examples in my book in the section on Fourier Series. Why is $\|f-s_n(f)\|_2=\inf_{T\in\mathcal{T}_n}\|f-T\|_2$? where $f$ is a continuous $2\pi$ periodic function, $T$ is ...
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1answer
66 views

Sum of $\sum_{n=1}^{\infty }\frac{1}{\pi n }\sin ^k\left(\frac{2\pi n}{k}\right)$

We have: $$S_k=\sum_{n=1}^{\infty }\frac{1}{\pi n }\sin ^k\left(\frac{2\pi n}{k}\right)$$ where $k$ is an odd number greater than $1$. I was able to find the sum of the series when $k=3,5$ as ...
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2answers
28 views

How does the Fourier transform get you the frequency amplitude

I understand that the Fourier transforn gets you the function which gives the amplitude of each frequency. But I don't understand how that is possible by multiplying it by an exponential. How is that ...
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1answer
64 views

wave equation on a square domain

I'm stuck on the following problem. Let $u(x, y, t)$ denote a solution to the linear wave equation $k^2(u_{xx}+u_{yy}) = u_{tt}$ with $k = 2$ on a square domain with corners at (0, 0), (0, 1), ...
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1answer
41 views

Fourier series for a logarithm

Is there an explicit Fourier sine series for the function $f$ defined below (valid for $x\in[0,\pi]$) ? $$f(x) := \ln\big(\sqrt{1 + \sin x} + \sqrt{\sin x}\big)$$ In case this is well known, a ...
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1answer
43 views

Find a non-Lipschitz Riemann integrable function that her Fourier series converge uniformly to her

This is my first question here. So I'll try to be short and to the point. I'm asked to find a Riemann integrable function $f$, that is not Lipschitz continuous but her Fourier series converge to $f$ ...
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0answers
20 views

Calculate the Fourier series with the trigonometry approach

I try to implement the Fourier series function in Python according to the following formulas: $Sf(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left [ a_{n}cos\frac{n\pi x}{L} + b_{n}sin\frac{n\pi x}{L} ...
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1answer
61 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
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1answer
27 views

A Fourier series' upper bound involving gamma function

I am reading Donald E. Knuth's "The Art of Computer Programming" Vol. 3 and stuck on the equation 47 and inequality 48 on Page 133,which are the follows: $$ \delta(n)=\frac{2}{\ln2}\sum_{k \ge 1} ...
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0answers
26 views

The functional $\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$

Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$. Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$. Let $G\triangleq\{\varphi\in ...
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1answer
53 views

dummy variable in Fourier transform confusion

In this text, why is it using different dummy variable for the integral of coefficients $a_n$ and $b_n$? I know that choosing the dummy variable does not affect integral but over here since we are ...
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1answer
33 views

Fourier coefficients, Fourier spectrum and signal energy

I was wondering if the equation 1) is correct in this example. The author described the difference between equations for $X(w)$ and $X[k]$. The only difference I see is the $\frac{1}{T}$ coefficient. ...
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0answers
44 views

Functions over a finite domain that cannot be represented by Fourier series

Given a double Fourier series for some $f:[0,L]\times [0,R]\to \mathbb{R}$ of the following form $$\sum_{k,l=0}^{\infty}a_{kl}\cos\left(\frac{2k\pi ...
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1answer
32 views

Expanding in a Fourier series $y = |\cos x|$

How to expanding in a Fourier series function $y = |\cos x|$? Especially interested in how to find $$a_n= \frac{2}{\pi}\int\limits_{0}^{\pi}|\cos x|\cos(nx)dx$$
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2answers
29 views

Explaining integrals equality

$$\int_{-2\pi}^0f(y)e^{iny} dy = \int_0^{2\pi}f(y)e^{-i(-n)y} dy $$ Can you please explain why is this equality true? I know that $\int_a^b f= - \int_b^a f$ but how is this applied here?
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1answer
41 views

Convergence of a complex Fourier series

Let $$ \sum_{k=-\infty}^\infty \frac{2}{\pi (2k+1)i} e^{(2k+1)it} $$ (*) For $n=2k$ the terms are zero. I'd be glad for a guidance. How do I approach this? Should I split it for Real/Imaginary?
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1answer
17 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
89 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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1answer
112 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
26 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
121 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
56 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
31 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 ...
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0answers
51 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
96 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
10 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
43 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
18 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
33 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
15 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\{ ...
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1answer
43 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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1answer
57 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
14 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
22 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!
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2answers
60 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
39 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 ...
2
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0answers
33 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
30 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
29 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
15 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
19 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
47 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
37 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
83 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
48 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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0answers
21 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
60 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
29 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
74 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
80 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 ...