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user 151496
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Prove that a symmetric filter preserve an p-order trend
Let the centered filter
$\Theta (L)=\theta _{-m}L^m+...+\theta _{-2}L^2+\theta _{-1}L+\theta _{0}+\theta _{1}L^{-1}+\theta _{2}L^{-2}+...+\theta _{m}L^{-m}$
where $\theta _{-m}+... … Under what conditions this filter preserves a p-order trend, that is $\Theta (L)t^p=t^p$?
So, applying the definition of the filter, we have
$\Theta (L)t^p=(\theta _{-m}L^m+... …
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