# Tagged Questions

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

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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### 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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### 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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### 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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### 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 ...