# Questions tagged [trigonometric-series]

For questions about or related to trigonometric series, i.e. series of the form $a_0 + \sum_{n = 1}^{\infty} (a_n \cos{nx} + b_n \sin{nx})$ or $\sum_n c_n e^{inx}$.

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### Show that $g(x)=\sum_{n \ge 1} \varepsilon_n b^{-n}\cos(b^n x)$ is in the Zygmund class $\lambda_*$ if $\varepsilon_n \rightarrow 0$

In his book Trigonometric Series, Zygmund says : Theorem : Let $b>1$, $\varepsilon_n \rightarrow 0$ and $$g(x)=\sum_{n \ge 1} \varepsilon_n b^{-n}\cos(b^n x)$$ then $g \in \lambda_*$ Which means ...
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### Sinusoids as solutions to differential equations

It is well known that the function $$t \mapsto a \cos(\omega t) + b \sin(\omega t)$$ is the solution to the differential equation: $$x''(t) = -\omega^2 x(t)$$ with the initial conditions $x(0) = a$ ...
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### How to write a square of a trigonometric polynomial cosine?

How to write a square of a polynomial of the form $$\left(1 + 2\sum_{k=1}^n a_k \cos k \theta\right)^2$$ with an explicit formula for just the coefficient of $$\cos k\theta$$ in terms of $k$ and the ...
### Showing $n=\sum_{k=1}^{(n+1)/2}\sin{\frac{2\pi k}{n+2}}\sin{\frac{\pi(n-2(k-1))}{n+2}}\sec^2{\frac{\pi(n-2(k-1))}{2n+4}}$ for natural $n$
Good afternoon, I am a little confused and intrigued by this finite summation formula I came up with. If $n$ is a natural number then  n=\sum_{k=1}^{(n+1)/2}\sin{\frac{2\pi k}{n+2}}\sin{\frac{\pi(n-...