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Trigonometric series

Ordinary Differerential Equations
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Papapetros Vaggelis
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Trigonometric series

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Post by Papapetros Vaggelis » Thu Feb 23, 2017 4:46 pm

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

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