  Let $u\left( t,y\right) $ be a polyharmonic function of order $N$ defined on
the strip $\left( a,b\right) \times\mathbb{R}^{d}$ satisfying the growth
condition $$ \sup_{t\in K}\left\vert u\left( t,y\right) \right\vert \leq
o\left( \left\vert y\right\vert ^{\left( 1-d\right)
/2}e^{\frac{\pi}{c}\left\vert y\right\vert }\right) $$ for $\left\vert
y\right\vert \rightarrow\infty$ and any compact subinterval $K$ of $\left(
a,b\right) $, and suppose that $u\left( t,y\right) $ vanishes on $2N-1$
equidistant hyperplanes of the form $\left\{ t_{j}\right\}
\times\mathbb{R}^{d}$ for $t_{j}=t_{0}+jc\in\left( a,b\right) $ and $j=-\left(
N-1\right) ,...,N-1.$ Then it is shown that $u\left( t,y\right) $ is odd at
$t_{0},$ i.e. that $u\left( t_{0}+t,y\right) =-u\left( t_{0}-t,y\right) $ for
$y\in\mathbb{R}^{d}$. The second main result states that $u$ is identically
zero provided that $u$ satisfies the growth condition and vanishes on $2N$
equidistant hyperplanes with distance $c.$
