A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

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 $\sup_{t\in K}\left\vert u\left( t,y\right) \right\vert \leq o\left( \left\vert y\right\vert ^{\left( 1-d\right) /2}e^{\fracπ{c}\left\vert y\right\vert }\right)$ 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 $c.$

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