# Questions tagged [fourier-series]

A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.

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### Error in solution of 3-45 in Solution Manual of Signals and Systems Oppenheim 2nd edition

Hello I was solving the question 3-45 of the book Signals and Systems 2nd edition and I assume there is an error in the solution manual, kindly if someone confirm that am I right? or the solution ...
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### Fourier series expansion of a $L^2$ function.

For a function $f \in L^2(\mathbb{T})$ (where $\mathbb{T}$ denotes the unit circle) I know that it can be expressed as $f(z) = \sum_{j = -\infty}^{\infty}f_j z^j$. The Fourier coefficients are given ...
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### Fourier series of $f(x) = \cos(x)$

The Fourier series of $f(x) = \cos(x)$ on $|x| < \pi/2$ and $f(x) = 0$ otherwise. Since $\cos(x) = \cos(-x)$, we have $b_n=0$. Then I computed 2 different $a_0$'s, thinking the latter one is the ...
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### Fourier series of $f(x) = 1$ on the interval $\pi/2 < |x| < \pi$

I am trying to calculate the Fourier series of $f(x) = 1$ on the interval $\pi/2 < |x| < \pi$ and $f(x) = 0$ otherwise. $f(x) = 1$ is an even function. Therefore, $b_n = 0$. I am troubling how ...
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### Can one show that $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$ for $a_k \in \mathbb{R}$?

This is a complex analysis puzzle that seems tricky. How can one show that for $a_k \in \mathbb{R}$, $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$? A hint is to consider ...
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