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## Trigonometric series

Ordinary Differerential Equations
Papapetros Vaggelis
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### Trigonometric series

Let $\displaystyle{\left(f_{m}\right)_{m\in\mathbb{Z}}}$ be a complex sequence such that

$\displaystyle{\sum_{m\in\mathbb{Z}}|f_m|<\infty}$. Consider the continuous function $\displaystyle{f:\left(-\pi,\pi\right]\to \mathbb{C}}$

defined by $\displaystyle{f(t)=\sum_{m\in\mathbb{Z}}f_m\,e^{i\,m\,t}}$ and the differential equation

$\displaystyle{x''+2\,k\,x'+d^2\,x=f}$ , where $\displaystyle{k\,,d}$ are positive constants.

Suppose a solution defined by $\displaystyle{x(t)=\sum_{m\in\mathbb{Z}}x_m\,e^{i\,m\,t}\,,-\pi<t\leq \pi}$

and find $\displaystyle{x_m\,,m\in\mathbb{Z}}$.

Note

The complex space $\displaystyle{\left(C((-\pi,\pi]),+\right)}$ has an inner product

$\displaystyle{\langle{f,g\rangle}=\dfrac{1}{2\,\pi}\,\int_{-\pi}^{\pi}f(t)\,\overline{g(t)}\,\mathrm{d}t\,\,\forall\,f\,,g\in C((-\pi,\pi])}$.